A Validation of the Phenomenon of Linearly Many Faults on Burnt Pancake Graphs with Its Applications

Abstract

“Linearly many faults” is a phenomenon observed by Cheng and Lipták in which a specific structure emerges when a graph is disconnected and often occurs in various interconnection networks. This phenomenon means that if a certain number of vertices or edges are deleted from a graph, the remaining part either stays connected or breaks into one large component along with smaller components with just a few vertices. This phenomenon can be observed in many types of graphs and has important implications for network analysis and optimization. In this paper, we first validate the phenomenon of linearly many faults for surviving graph of a burnt pancake graph $B{P}_n$ when removing any edge subset with a size of approximately six times $\lambda \left(B{P}_n\right)$. For graph G, the ℓ-component edge connectivity denoted as ${\lambda }_{\ell }\left(G\right)$ (resp., the ℓ-extra edge connectivity denoted as ${\lambda }^{\left(\ell \right)}\left(G\right)$) is the cardinality of a minimum edge subset S such that $G-S$ is disconnected and has at least ℓ components (resp., each component of $G-S$ has at least $\ell +1$ vertices). Both ${\lambda }_{\ell }\left(G\right)$ and e${\lambda }^{\left(\ell \right)}\left(G\right)$ are essential metrics for network reliability assessment. Specifically, from the property of “linearly many faults”, we may further prove that ${\lambda }_5\left(B{P}_n\right)={\lambda }^{\left(3\right)}\left(B{P}_n\right)+3=4n-3$ for $n⩾5$; ${\lambda }_6\left(B{P}_n\right)={\lambda }^{\left(4\right)}\left(B{P}_n\right)+4=5n-4$ and ${\lambda }_7\left(B{P}_n\right)={\lambda }^{\left(5\right)}\left(B{P}_n\right)+5=6n-5$ for $n⩾6$.

1. Introduction

Investigating interconnection networks and their intrinsic properties is crucial for developing efficient parallel and distributed computer systems. A simple undirected graph represents the underlying topology of such a system called an interconnection network, where vertices represent a processor, and edges represent communication links between processors. Therefore, a well-structured network topology can lead to higher benefits for the system operation, including fault-tolerant data transmission and system reliability. For convenience, the terms graphs and networks are used interchangeably.

1.1. Background

It is almost impossible to design a multiprocessor system without defects. Connectivity $\kappa \left(G\right)$ and edge connectivity $\lambda \left(G\right)$ are used to adjudicate a network’s reliability and fault tolerance. Usually, a fundamental property of an interconnection network is that it must possess regularity (i.e., every vertex in the network has the same degree). In particular, it is better if it meets the maximal connectivity (i.e., the connectivity equals the regularity of the graph). It is interesting to think about what would happen if we were to remove more than n vertices or edges from an n-regular graph. In such cases, two possible scenarios could arise. Either the resulting graph would remain connected, or it would split into several components, with the smaller component containing just a singleton. Further, one may wonder what exactly would happen if about $2n,3n,4n,5n,\dots$ or more vertices or edges were further removed. When multiple failures occur at the same time and the graph becomes disconnected, the best-case scenario is when a large component containing most of the remaining vertices is retained, along with some smaller components. This way, the subnetwork represented by the large component can continue to function effectively. In fact, this phenomenon of a disconnected graph caused by failures was first discovered in the pioneering work of Yang et al. [1]. Later, Cheng and Lipt’ak [2] popularized this concept and formally called this phenomenon the “linear many faults” property. Since then, this property has attracted much attention in the research for other networks, e.g., Cayley graphs generated by transposition trees [2], 2-tree [3], transposition triangle free unicyclic graphs [4], $(n,k)$-star graphs [5], arrangement graphs [6], augmented cubes [7], and dual-cube-like networks [8]. Mainly, this property can export network metrics related to fault tolerance [9,10,11,12].

Regarding Cayley graphs generated by transposition trees, we let $\mathsf{\Gamma }$ be a finite group and S a subset of $\mathsf{\Gamma }$. The Cayley digraph of Γ generated by S, denoted by Cay$(\mathsf{\Gamma },S)$, is digraph with vertex set $\mathsf{\Gamma }$ and arc set $\left\{\right(\gamma ,\gamma s\left)\right|\gamma \in \mathsf{\Gamma }\mathrm{and}s\in S\}$. If S does not include the identity and $S=S^{-1}=\left\{s^{-1}\right|s\in S\}$, then Cay$(\mathsf{\Gamma },S)$ is an undirected simple graph. We let $\left[n\right]=\{1,2,\dots ,n\}$, $Sym\left(n\right)$ be the symmetric group on $\left[n\right]$, and T be a set of transpositions of $Sym\left(n\right)$. Then, Cay$\left(Sym\right(n),T)$ is called the Cayley graphs generated by transposition tree T if $G\left(T\right)$ is a tree with the vertex set $\left[n\right]$ such that edge $uv\in E\left(G\right(T\left)\right)$ if and only if the corresponding transposition $\left(uv\right)\in T$.

To better understand the reliability of networks, Harary [13] proposed a concept called conditional connectivity which involves attaching certain conditions to connected components. Additionally, Fábrega and Fiol [14] introduced two generalizations of classical connectivity, namely extra connectivity and extra edge connectivity, which help to ensure the scale of each component. Later on, Sampathkumar [15] and Chartrand et al. [16] independently introduced a generalization of classical (edge) connectivity regarding the number of components for disconnected graphs, the former called general connectivity and the latter called generalized connectivity. Henceforth, we adopt appropriate terms called component connectivity and component edge connectivity, suggested by Hsu et al. [17] and Zhao et al. [18], respectively. For the recent results of interconnection networks, please refer to [19,20,21,22,23] for extra (edge) connectivity, [24,25,26,27,28,29] for component (edge) connectivity, and [12,30,31,32] for relationship between these two kinds of (edge) connectivity. In addition, for research on connectivity related to diverse graph indices (such as the Wiener index, the Zagreb index, the Randic index, etc.) with fuzzy information and their applications, please refer to [33,34,35,36].

This paper investigates the “linear many faults” property on a burnt pancake graph $B{P}_n$, which is the Cayley graph of the group of signed permutations generated by prefix reversals and defined by Gates and Papadimitriou in 1979 [37]. $B{P}_n$ attracts the attention of researchers mainly because of another accompanying interesting definition called the pancake graph, which refers to the mathematic puzzle of sorting a pile of unordered pancakes in the size order. In this case, a spatula could be inserted anywhere in the pancake stack to flip all the pancakes above it. The minimum number of flips required to sort the given pancakes is called the pancake number. Hence, the operation of flips is called the prefix reversal when we treat the stack of pancakes as a sequence of symbols, and acquiring the pancake number is equal to obtaining the diameter of the pancake graph. Then, $B{P}_n$ introduces the change in positive and negative signs, making this question more interesting. However, there has yet to be a general solution to the diameter problem of these two classes of graphs so far [38].

For burnt pancake graphs, the earliest research mainly pursued their diameters, while the current research focuses on exploring fault tolerance [39,40] and diagnosis [40,41]. In addition, many diverse connectivities have been investigated in the literature, including spanning connectivity [42], structure connectivity [43], neighbor connectivity [44,45], and component connectivity [25,28]. Following the direction of probing connectivity, this paper first proves that when removing any edge subset with a size of approximately six times $\lambda \left(B{P}_n\right)$, the surviving graph possesses the “linearly many faults” property. According to this characteristic, we obtain component edge connectivity and extra edge connectivity of $B{P}_n$ for certain dimensions n, extending the results of [30]. Specifically, we prove that ${\lambda }_5\left(B{P}_n\right)={\lambda }^{\left(3\right)}\left(B{P}_n\right)+3=4n-3$ for $n⩾5$; ${\lambda }_6\left(B{P}_n\right)={\lambda }^{\left(4\right)}\left(B{P}_n\right)+4=5n-4$ and ${\lambda }_7\left(B{P}_n\right)={\lambda }^{\left(5\right)}\left(B{P}_n\right)+5=6n-5$ for $n⩾6$.

1.2. Organization

Section 2 introduces definitions and necessary terminologies and notations. Also, burnt pancake graphs and related properties are given. Section 3 shows the existence of the “linearly many faults” property for the surviving graph of $B{P}_n$ when the removal of an edge subset with a size of approximately six times $\lambda \left(B{P}_n\right)$. Section 4 obtains some relations between component edge connectivity and extra edge connectivity of $B{P}_n$ through the derived property. Finally, we add concluding remarks in Section 5.

2. Preliminaries

2.1. Definitions and Terminologies

Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Two vertices u and v are adjacent if they are joined by an edge, where u and v are called neighbors to each other. For vertex $u\in V\left(G\right)$, let $N_G\left(u\right)$ be the set of neighbors of u in G. For $U\subseteq V\left(G\right)$, the open neighborhood of U in G is defined as $N_G\left(U\right)={\cup }_{u\in U}{N}_G\left(u\right)-U$. The edge neighborhood of U in G, denoted as $N_{E\left(G\right)}\left(U\right)$ (or $N_E\left(U\right)$), is the set of edges incident with at least one vertex of U in G. Also, denote $G\left[U\right]$ the subgraph of G induced by U. For two disjoint subgraphs (or vertex sets) $H_1$ and $H_2$, let $E(H_1,H_2)$ be the set of edges with one end in $H_1$ and the other in $H_2$. A cycle (resp., path) of length k is called a k-cycle (resp., k-path), denoted by $C_k$ (resp., $P_k$).

Let G be a graph. The connectivity (resp., edge connectivity) of G, denoted by $\kappa \left(G\right)$ (resp., $\lambda \left(G\right)$), is the minimum number of vertices (resp., edges) that need to be removed to disconnect G or become a trivial graph. For $S\subseteq V\left(G\right)$ (resp., $S\subseteq E\left(G\right)$), let $G-S$ be the graph that removes vertices (resp., edges) of S from G. Particularly, S is a vertex-cut (resp., edge-cut) of G provided $G-S$ is disconnected. In $G-S$, the component with the largest number of vertices is called the large component, and a component that is not the largest one is called the smaller component.

Graph G is super h-vertex-connected (resp., super h-edge-connected) of order q if, after deleting at most h vertices (resp., h edges), the resulting graph is either connected or has one large component along with smaller components containing totally at most q vertices. In other words, the resulting graph has a component of size at least $\left|V\right(G-F\left)\right|-q$ with $\left|F\right|⩽h$. The following result is helpful throughout the paper.

Proposition 1

([9]). Let $q⩾1$ be an integer. If a connected graph G with at least $max\{m+2q+4,3q+1\}$ vertices is super-m-vertex-connected of order q, then G is super-m-edge-connected of order q.

Definition 1

(see [17]). Let G be a connected graph and $F\subset E\left(G\right)$. If $G-F$ is disconnected and has at least ℓ components, then F is called anℓ-component edge-cut. The ℓ-component edge connectivity of G, denoted by ${\lambda }_{\ell }\left(G\right)$, is the cardinality of a minimum ℓ-component edge-cut of G. Obviously, ${\lambda }_{\ell +1}\left(G\right)⩾{\lambda }_{\ell }\left(G\right)$ and ${\lambda }_2\left(G\right)=\lambda \left(G\right)$ for every positive integer ℓ.

Definition 2

(see [14]). Let G be a connected graph and $F\subset E\left(G\right)$. If $G-F$ is disconnected and every component of $G-F$ has at least $h+1$ vertices, then F is called an h-extra edge-cut. The h-extra edge connectivity of G, denoted by ${\lambda }^{\left(h\right)}\left(G\right)$, is the cardinality of a minimum h-extra edge-cut, if it exists. Obviously, ${\lambda }^{(h+1)}\left(G\right)⩾{\lambda }^{\left(h\right)}\left(G\right)$ and ${\lambda }^{\left(0\right)}\left(G\right)=\lambda \left(G\right)$.

Lemma 1

(see [30]). Let H be a connected graph and $k<\left|V\right(H\left)\right|/2$ be an integer. Let

$$X^{*}=arg\underset{X\subseteq V\left(H\right)}{min}\left\{\right|E(X,H-X)|:|X|=k,H\left[X\right]\mathrm{and}H-X\mathrm{are}\mathrm{connected}\mathrm{subgraphs}\},$$

$h=\left|E\right(X^{*},H-X^{*}\left)\right|$, and $\ell =\left|E\right(H\left[X^{*}\right]\left)\right|$. If H fulfills the following:

(i)

For $F\subseteq E\left(H\right)$ with $\left|F\right|⩽h-1$, $H-F$ has a large component along with small components containing totally at most $k-1$ vertices;

(ii)

For $F^{\prime }\subseteq E\left(H\right)$ with $|F^{\prime }|⩽h+\ell -1$, $H-F^{\prime }$ has at most k components;

then ${\lambda }_{k+1}\left(H\right)=h+\ell ={\lambda }^{(k-1)}\left(H\right)+\ell$.

2.2. Burnt Pancake Graphs $B{P}_n$

We put a negative sign on the top of a symbol for notational convenience, e.g., $\overline{k}=-k$. We let $\left[n\right]=\{1,2,\dots ,n\}$ and $\langle n\rangle =\left[n\right]\cup \{\overline{k}:k\in \left[n\right]\}$. A signed permutation of $\left[n\right]$ is an permutation $x_1{x}_2\cdots x_n$ of $\langle n\rangle$ such that $|x_1\left|\right|x_2|\cdots |x_n|$ (each element takes the absolute value) forms a permutation of $\left[n\right]$. For signed permutation $x=x_1{x}_2\cdots x_i\cdots x_n$ of $\langle n\rangle$ and integer $i\in \left[n\right]$, the ith prefix reversal of x is defined by $x^i={\overline{x}}_i{\overline{x}}_{i-1}\cdots {\overline{x}}_1{x}_{i+1}\cdots x_n$.

Definition 3

(see [37]). An n-regular graph $B{P}_n$ with $n!2^n$ vertices is called the n-dimensional burnt pancake network if every vertex of $B{P}_n$ has a unique label from the signed permutation of $\langle n\rangle$ such that $uv\in E\left(B{P}_n\right)$ if and only if $u^i=v$ for $i\in \left[n\right]$. The edge $uv$ is called an i-dimensional edge and u is called the i-neighbor of v, and vice versa.

Figure 1 depicts $B{P}_n$ for all $n\in \left[3\right]$, where we use different types of line to draw distinct dimensional edges. Clearly, every vertex of $B{P}_n$ has a unique k-neighbor for $k\in \left[n\right]$. By definition, $B{P}_n$ is decomposed into $2n$ vertex-disjoint subgraphs $B{P}_n^k$ for $k\in \langle n\rangle$ such that every vertex in a subgraph fixes the symbol k in the rightmost position. Clearly, $B{P}_n^k$ is isomorphic to $B{P}_{n-1}$. An external edge of $B{P}_n$ is one whose two ends are in distinct $B{P}_n^k$s. For $u\in V\left(B{P}_n^i\right)$, the unique neighbor outside $B{P}_n^k$ is called the external neighbor of u. Indeed, an external edge is an n-dimensional edge. Also, $E_{j,k}\left(B{P}_n\right)$ denotes the set of edges between $B{P}_n^j$ and $B{P}_n^j$ for $j,k\in \langle n\rangle$ with $j\ne k$.

Lemma 2

(see [39,42,46]). For $B{P}_n$, the following properties hold:

(1)

$B{P}_n$ is an n-regular graph with $n\times 2^{n-1}\times n!$ edges. $|E_{j,k}\left(B{P}_n\right)|=2^{n-2}\times (n-2)!$ if $j\ne \overline{k}$, and $|E_{j,\overline{k}}\left(B{P}_n\right)|=0$.

(2)

For $n⩾2$, $\kappa \left(B{P}_n\right)=\lambda \left(B{P}_n\right)=n$.

(3)

For $n⩾2$, the girth of $B{P}_n$ is $g\left(B{P}_n\right)=8$.

Lemma 3

(see [40,41]). For $n⩾4$, we let F be a vertex-cut of $B{P}_n$. The following properties hold:

(1)

If $\left|F\right|⩽2n-2$, $B{P}_n-F$ has two components, one of which is a singleton or an edge. Furthermore, if the small component is an edge, then F is the neighborhood of this edge and $\left|F\right|=2n-2$.

(2)

If $\left|F\right|⩽3n-5$, $B{P}_n-F$ has a large component along with smaller components containing totally at most two vertices.

(3)

If $\left|F\right|⩽4n-7$, $B{P}_n-F$ has a large component along with smaller components containing totally at most three vertices.

Lemma 4

(see [28]). For $n⩾5$, we let F be a vertex-cut of $B{P}_n$. If $\left|F\right|⩽5n-9$, $B{P}_n-F$ contains a large component along with smaller components containing totally at most four vertices.

Lemma 5.

For $n⩾4$, we let F be an edge-cut of $B{P}_n$. The following properties hold:

(1)

If $\left|F\right|⩽2n-2$, $B{P}_n-F$ has two components, one of which is a singleton or an edge. Furthermore, if the small component is an edge, then F is the neighborhood of this edge and $\left|F\right|=2n-2$.

(2)

If $\left|F\right|⩽3n-5$, $B{P}_n-F$ has a large component along with smaller components containing totally at most two vertices.

(3)

If $\left|F\right|⩽4n-7$, $B{P}_n-F$ has a large component along with smaller components containing totally at most three vertices.

Proof. 

By Lemma 3, $B{P}_n$ is super-$(2n-3)$-vertex-connected of order 1, $(3n-5)$-vertex-connected of order 2, and $(4n-7)$-vertex-connected of order 3, respectively. We note that $|V\left(B{P}_n\right)|=n!2^n>max\{(2n-3)+2\times 1+4,3\times 1+1\}$ (resp., $n!2^n>max\{(3n-5)+2\times 2+4,3\times 2+1\}$ and $n!2^n>max\{(4n-7)+2\times 3+4,3\times 3+1\}$) for $n\ge 4$. By Proposition 1, $B{P}_n$ is super-$(2n-3)$-edge-connected of order 1, $(3n-5)$-edge-connected of order 2, and $(4n-7)$-edge-connected of order 3, respectively. Thus, the lemma follows. □

Lemma 6.

For $n⩾5$, we let F be an edge-cut of $B{P}_n$. If $\left|F\right|⩽5n-9$, $B{P}_n-F$ has a large component along with smaller components containing totally at most four vertices.

Proof. 

By Lemma 4, $B{P}_n$ is super-$(5n-9)$-vertex-connected of order 4. We note that $|V\left(B{P}_n\right)|=n!2^n>max\{(5n-9)+2\times 4+4,3\times 4+1\}=max\{5n+3,13\}$ for $n\ge 5$. By Proposition 1, $B{P}_n$ is super-$(5n-9)$-edge-connected of order 4, and the result holds. □

3. Linearly Many Faults in Burnt Pancake Graphs

In this section, we focus on the linearly many faults in burnt pancake graphs.

Lemma 7.

For $B{P}_n$ with $n⩾4$ and $X\subset V\left(B{P}_n\right)$, if $\left|X\right|=4$, then $\left|E\right(X,B{P}_n-X\left)\right|⩾4n-6$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|⩾4n-3$.

Proof. 

Let $X=\{u,v,x,y\}$. Note that $B{P}_n$ has no k-cycle for $k⩽4$. By Lemma 2, $\kappa \left(B{P}_n\right)=\lambda \left(B{P}_n\right)=n$ and $g\left(B{P}_n\right)=8$.

If $B{P}_n\left[X\right]$ contains four singletons, then $|E(X,B{P}_n-X)|=|N_{E\left(B{P}_n\right)}\left(X\right)|=4n$.

If $B{P}_n\left[X\right]$ contains two singletons and an edge, then $|E(X,B{P}_n-X)|=2n+2(n-1)=4n-2$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=4n-2+1=4n-1$.

If $B{P}_n\left[X\right]$ contains (i) two edges or (ii) a 2-path and a singleton, then $|E(X,B{P}_n-X)|=2(n-1)+2(n-1)=4n-4$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=4n-4+2=4n-2$.

If $B{P}_n\left[X\right]$ contains (i) a 3-path or (ii) a graph isomorphic to $K_{1,3}$, then $|E(X,B{P}_n-X)|=2(n-1)+2(n-2)=4n-6$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=4n-6+3=4n-3$.

Table 1. All cases of $B{P}_n\left[X\right]$ for $X=\{x,y,u,v\}$.

	${\mathit{BP}}_{\mathit{n}}\left[\mathit{X}\right]$	$\mathit{E}(\mathit{X},{\mathit{BP}}_{\mathit{n}}-\mathit{X})$	${\mathit{N}}_{\mathit{E}\left({\mathit{BP}}_{\mathit{n}}\right)}\left(\mathit{X}\right)$
1	four singletons	$4n$	$4n$
2	an edge and two singletons	$4n-2$	$4n-1$
3	a 2-path and a singleton	$4n-4$	$4n-2$
4	two edges	$4n-4$	$4n-2$
5	a graph isomorphic to $K_{1,3}$	$4n-6$	$4n-3$
6	a 3-path	$4n-6$	$4n-3$

lists all cases of $B{P}_n\left[X\right]$. Hence, $\left|E\right(X,B{P}_n-X\left)\right|⩾4n-6$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|⩾4n-3$. □

Lemma 8.

For $B{P}_n$ with $n⩾4$ and $X\subset V\left(B{P}_n\right)$, if $\left|X\right|=5$, then $\left|E\right(X,B{P}_n-X\left)\right|⩾5n-8$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|⩾5n-4$.

Proof. 

Let $X=\{x,y,z,u,v\}$. Note that $B{P}_n$ has no k-cycle for $k⩽5$. By Lemma 2, $\kappa \left(B{P}_n\right)=\lambda \left(B{P}_n\right)=n$ and $g\left(B{P}_n\right)=8$.

If $B{P}_n\left[X\right]$ contains five singletons, then $|E(X,B{P}_n-X)|=|N_{E\left(B{P}_n\right)}\left(X\right)|=5n$.

If $B{P}_n\left[X\right]$ contains three singletons and an edge, then $|E(X,B{P}_n-X)|=3n+2(n-1)=5n-2$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=5n-2+1=5n-1$.

If $B{P}_n\left[X\right]$ contains (i) two edges and a singleton or (ii) a 2-path and two singletons, then $|E(X,B{P}_n-X)|=2(n-1)+2(n-1)+n=5n-4$ (resp., $|E(X,B{P}_n-X)|=2(n-1)+(n-2)+2n=5n-4$) and $|N_{E\left(B{P}_n\right)}\left(X\right)|=5n-4+2=5n-2$.

If $B{P}_n\left[X\right]$ contains (i) a 2-path and an edge, (ii) a 3-path and a singleton, or (iii) a graph isomorphic to $K_{1,3}$ and a singleton, then $|E(X,B{P}_n-X)|=2(n-1)+2(n-1)+n-2=5n-6$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=5n-6+3=5n-3$.

If $B{P}_n\left[X\right]$ contains (i) a 4-path, (ii) a graph isomorphic to $K_{1,4}$, or (iii) a tree with five vertices, then $|E(X,B{P}_n-X)|=2(n-1)+3(n-2)=5n-8$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=5n-8+4=5n-4$.

All cases of the induced subgraph $B{P}_n\left[X\right]$ are listed in

Table 2. All cases of $B{P}_n\left[X\right]$ for $X=\{x,y,z,u,v\}$.

	${\mathit{BP}}_{\mathit{n}}\left[\mathit{X}\right]$	$\mathit{E}(\mathit{X},{\mathit{BP}}_{\mathit{n}}-\mathit{X})$	${\mathit{N}}_{\mathit{E}\left({\mathit{BP}}_{\mathit{n}}\right)}\left(\mathit{X}\right)$
1	five singletons	$5n$	$5n$
2	an edge and three singletons	$5n-2$	$5n-1$
3	two edges and a singleton	$5n-4$	$5n-2$
4	a 2-path and two singletons	$5n-4$	$5n-2$
5	a 2-path and an edge	$5n-6$	$5n-3$
6	a 3-path and a singleton	$5n-6$	$5n-3$
7	a graph isomorphic to $K_{1,3}$ and a singleton	$5n-6$	$5n-3$
8	a 4-path, Figure 2a	$5n-8$	$5n-4$
9	a graph isomorphic to $K_{1,4}$, Figure 2c	$5n-8$	$5n-4$
10	a tree with 5 vertices, Figure 2b	$5n-8$	$5n-4$

(see Figure 2a–c). Hence, $\left|E\right(X,B{P}_n-X\left)\right|⩾5n-8$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|⩾5n-4$. □

Lemma 9.

For $B{P}_n$ with $n⩾4$ and $X\subset V\left(B{P}_n\right)$, if $\left|X\right|=6$, then $\left|E\right(X,B{P}_n-X\left)\right|⩾6n-10$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|⩾6n-5$.

Proof. 

Let $X=\{x,y,z,u,v,w\}$. Note that $B{P}_n$ has no k-cycle for $k⩽6$. By Lemma 2, $\kappa \left(B{P}_n\right)=\lambda \left(B{P}_n\right)=n$ and $g\left(B{P}_n\right)=8$.

If $B{P}_n\left[X\right]$ contains six singletons, then $|E(X,B{P}_n-X)|=|N_{E\left(B{P}_n\right)}\left(X\right)|=6n$.

If $B{P}_n\left[X\right]$ contains four singletons and an edge, then $|E(X,B{P}_n-X)|=4n+2(n-1)=6n-2$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=6n-2+1=6n-1$.

If $B{P}_n\left[X\right]$ contains (i) two edges and two singleton or (ii) a 2-path and three singletons, then $|E(X,B{P}_n-X)|=2(n-1)+2(n-1)+2n=6n-4$ (resp., $|E(X,B{P}_n-X)|=2(n-1)+(n-2)+3n=6n-4$) and $|N_{E\left(B{P}_n\right)}\left(X\right)|=6n-4+2=6n-2$.

If $B{P}_n\left[X\right]$ contains (i) a 3-path and two singletons, (ii) three edges, or (iii) an edge, a 2-path and a singleton, then $|E(X,B{P}_n-X)|=2(n-1)+2(n-2)+2n=6n-6$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=6n-6+3=6n-3$.

If $B{P}_n\left[X\right]$ contains (i) a singleton and a tree with five vertices Figure 2a–c, (ii) an edge and a 3-path or a graph isomorphic to $K_{1,3}$, or (iii) two 2-paths, then $|E(X,B{P}_n-X)|=2(n-1)+2(n-1)+2(n-2)=6n-8$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=6n-8+4=6n-4$.

If $B{P}_n\left[X\right]$ contains (i) a 5-path, (ii) a graph isomorphic to $K_{1,5}$, or (iii) a tree with 6 vertices, isomorphic to one of Figure 3b–d, then $|E(X,B{P}_n-X)|=2(n-1)+4(n-2)=6n-10$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|=6n-10+5=5n-5$.

All cases of the induced subgraph $B{P}_n\left[X\right]$ are listed in

Table 3. All cases of $B{P}_n\left[X\right]$ for $X=\{x,y,z,u,v,w\}$.

	${\mathit{BP}}_{\mathit{n}}\left[\mathit{X}\right]$	$\mathit{E}(\mathit{X},{\mathit{BP}}_{\mathit{n}}-\mathit{X})$	${\mathit{N}}_{\mathit{E}\left({\mathit{BP}}_{\mathit{n}}\right)}\left(\mathit{X}\right)$
1	six singletons	$6n$	$6n$
2	an edge and four singletons	$6n-2$	$6n-1$
3	a 2-path and three singletons	$6n-4$	$6n-2$
4	a 3-path and two singletons	$6n-6$	$6n-3$
5	two singletons and two edges	$6n-4$	$6n-2$
6	a 4-path and a singleton	$6n-8$	$6n-4$
7	a singleton and a graph isomorphic to $K_{1,4}$	$6n-8$	$6n-4$
8	a singleton and a tree with 5 vertices, Figure 2b	$6n-8$	$6n-4$
9	three edges	$6n-6$	$6n-3$
10	an edge and 3-path	$6n-8$	$6n-4$
11	an edge and a graph isomorphic to $K_{1,3}$	$6n-8$	$6n-4$
12	an edge, a singleton and 2-path	$6n-6$	$6n-3$
13	two 2-paths	$6n-8$	$6n-4$
14	a 5-path, Figure 3a	$6n-10$	$6n-5$
15	a graph isomorphic to $K_{1,5}$, Figure 3f	$6n-10$	$6n-5$
16	a tree with 6 vertices, isomorphic to one of Figure 3b–d	$6n-10$	$6n-5$

(see Figure 3). Hence, $\left|E\right(X,B{P}_n-X\left)\right|⩾6n-10$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|⩾6n-5$. □

Figure 3. (a–f) Six trees with six vertices.

Figure 3. (a–f) Six trees with six vertices.

Lemma 10.

For $B{P}_n$ with $n⩾4$ and $W\subset V\left(B{P}_n\right)$, if $\left|W\right|=7$, then $\left|E\right(W,B{P}_n-W\left)\right|⩾7n-12$ and $|N_{E\left(B{P}_n\right)}\left(W\right)|⩾7n-6$.

Proof. 

By Lemma 2, we have $g\left(B{P}_n\right)=8$. We let H be a connected subgraph of $B{P}_n$ that does not contain a 6-path. Then, any vertex $x\in V\left(B{P}_n\right)\setminus V\left(H\right)$ can connect to at most one vertex in H; otherwise, the subgraph induced by $V\left(H\right)\cup \left\{x\right\}$ produces a cycle of length of less than 8. Particularly, we consider H a component of $B{P}_n\left[X\right]$, where X is a subset of $V\left(B{P}_n\right)$ with $\left|X\right|=6$ shown in Table 3. We let t be the number of components of $B{P}_n\left[X\right]$ and let $W=X\cup \left\{x\right\}$, where $x\in V\left(B{P}_n\right)\setminus X$. Clearly, $|E(x,B{P}_n-\left\{x\right\})|=|N_{E\left(B{P}_n\right)}\left(x\right)|=n$ and, from the above reasoning, x may connect to at most t vertices of X in $B{P}_n$, i.e., $\left|E\right(\left\{x\right\},X\left)\right|⩽t$. By checking all sixteen cases in Table 3, we have $|E(X,B{P}_n-X)|-2|E(\left\{x\right\},X)|⩾|E(X,B{P}_n-X)|-2t⩾6n-12$ and $|N_{E\left(B{P}_n\right)}\left(X\right)|-|E(\left\{x\right\},X)|⩾|N_{E\left(B{P}_n\right)}\left(X\right)|-t⩾6n-6$. Thus,

$$\begin{array}{ccc}\hfill \left|E\right(W,B{P}_n-W\left)\right|& =& \left|E\right(X\cup \left\{x\right\},B{P}_n-(X\cup \left\{x\right\})\left)\right|\hfill \\ & =& |E(x,B{P}_n-\left\{x\right\})|+|E(U,B{P}_n-X)|-2|E(\left\{x\right\},X)|\hfill \\ & ⩾& n+(6n-12)\hfill \\ & =& 7n-12\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill |N_{E\left(B{P}_n\right)}\left(W\right)|& =& |N_{E\left(B{P}_n\right)}(X\cup \left\{x\right\})|\hfill \\ & =& |N_{E\left(B{P}_n\right)}\left(x\right)|+|N_{E\left(B{P}_n\right)}\left(X\right)|-|E(\left\{x\right\},X)|\hfill \\ & ⩾& n+(6n-6)\hfill \\ & =& 7n-6,\hfill \end{array}$$

as desired. □

We recall that $B{P}_n$ is decomposed into $2n$ vertex-disjoint subgraphs $B{P}_n^i$ for $i\in \langle n\rangle$ by fixing symbol i in the rightmost position for each vertex where each $B{P}_n^i$ is isomorphic to $B{P}_{n-1}$. Henceforth, we consider F to be an edge-cut of $B{P}_n$ and let $F_i=F\cap E\left(B{P}_n^i\right)$ and $f_i=\left|F_i\right|$ for each $i\in \langle n\rangle$. We let $F_c=F-{\displaystyle \sum _{i\in \langle n\rangle }}{F}_i$ and $f_c=\left|F_c\right|$. We let $I=\{i\in \langle n\rangle :f_i⩾n-1\}$ and $J=\langle n\rangle \setminus I$. Also, we define

$$F_I=\bigcup _{i\in I}{F}_i,F_J=\bigcup _{j\in J}{F}_j,f_I=|F_I|,f_J=\left|F_J\right|,\mathrm{and}B{P}_n^J=B{P}_n[\bigcup _{j\in J}V\left(B{P}_n^j\right)].$$

Theorem 1.

For $n⩾5$, we let $B{P}_n$ be the n-dimensional burnt pancake graph and $F\subset E\left(B{P}_n\right)$ be an arbitrary edge set. If $\left|F\right|⩽6n-11$, then $B{P}_n-F$ either is connected to or contains a large component along with smaller components containing totally at most five vertices.

Proof. 

We suppose that $B{P}_n-F$ is disconnected and let M be the union of smaller components of $B{P}_n-F$. By the definition of M, it suffices to show that $\left|V\right(M\left)\right|⩽5$. Since $\left|F\right|⩽6n-11$ and $f_i⩾n-1$ for $i\in I$, we have $\left|I\right|⩽5$. Then, $\left|J\right|=2n-\left|I\right|⩾2n-5⩾5$ when $n⩾5$. For each $j\in J$, as each subgraph $B{P}_n^j$ is isomorphic to $B{P}_{n-1}$, by Lemma 2(2), we have $f_j<n-1=\lambda \left(B{P}_{n-1}\right)$, and thus $B{P}_n^j-F_j$ is connected. We claim that the following remark holds.

Remark 1.

$B{P}_n^J-F_J$ is connected.

For $j,k\in J$ and $j\ne \overline{k}$, by Lemma 2(1), we have $|E_{j,k}\left(B{P}_n\right)|=(n-2)!\times 2^{n-2}>6n-11$ when $n⩾5$. Thus, $B{P}_n^j-F_j$ is connected with $B{P}_n^k-F_k$ through an external edge. Moreover, since $\left|J\right|⩾5$, if $k,\overline{k}\in J$, there exists $j\in J\setminus \{k,\overline{k}\}$ such that $B{P}_n^j-F_j$ is connected to each of $B{P}_n^k-F_k$ and $B{P}_n^{\overline{k}}-F_{\overline{k}}$ through external edges. Therefore, $B{P}_n^J-F_J$ is connected.

We prove the theorem by induction on n, and the proof is separated into two parts: Part I for base case ($n=5$) and Part II for induction step ($n⩾6$).

For base case, if $n=5$, then $\left|F\right|⩽6n-11=19$. We note that $B{P}_5$ can be decomposed into 10 vertex-disjoint subgraphs, denoted by $B{P}_5^i$, by fixing symbol i in the rightmost position of each vertex for $i\in \langle 5\rangle$. Obviously, $B{P}_5^i$ is isomorphic to $B{P}_4$. As $I=\{i\in \langle 5\rangle :f_i⩾n-1=4\}$, we have $\left|I\right|⩽4$; otherwise, $\left|F\right|⩾(n-1)\left|I\right|⩾4\times 5>19$. By Remark 1, $B{P}_5^J-F_J$ is connected. If $\left|I\right|=0$, then $B{P}_5-F=B{P}_5^J-F_J$ is connected; the result holds. We now consider $1⩽\left|I\right|⩽4$. For each $i\in I$, we let $S_i\subset V\left(B{P}_n^i\right)$ be the set of vertices that do not belong to the large component of $B{P}_n^i-F_i$. We consider the following cases.

Case I-1. $\left|I\right|=1$. We let $I=\left\{i\right\}$. For $4⩽f_i⩽11=5(n-1)-9$, if $B{P}_5^i-F_i$ is disconnected, by Lemma 6, it has a large component and is with $\left|S_i\right|⩽4$. Since every vertex of $B{P}_5^i$ has an external edge, there are $2^{n-1}(n-1)!=4!\times 2^4$ edges between $B{P}_5^i$ and $B{P}_5^J-F_J$. Also, since $4!\times 2^4>19⩾\left|F\right|$, the large component of $B{P}_5^i-F_i$ is connected to $B{P}_5^J-F_J$. This implies that $\left|V\left(M\right)\right|⩽|S_i|⩽4$ (see Figure 4a).

It remains to consider $12⩽f_i⩽19$. In this case, we have $f_c⩽\left|F\right|-f_i⩽19-12=7$. Since every vertex of M has exactly one external neighbor, we have $\left|V\left(M\right)\right|⩽f_c⩽7$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_5-V\left(M\right)\left)\right|⩾6n-10=20>19$, a contradiction. Similarly, if $\left|V\right(M\left)\right|=7$, by Lemma 10, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾7n-12=23>19$, a contradiction. This implies that $\left|V\right(M\left)\right|⩽5$ (see Figure 4b).

Case I-2. $\left|I\right|=2$. We let $I=\{i,j\}$ and, without loss of generality, we suppose $f_i⩽f_j$. Since $\left|F\right|⩽19$, we have $4⩽f_i⩽9$; otherwise, $f_i+f_j⩾2f_i⩾20$. We first consider $4⩽f_j⩽9$. For each $\ell \in I$, as $f_{\ell }⩽9=4(n-1)-7$, by Lemma 5(3), if $B{P}_5^{\ell }-F_{\ell }$ is disconnected, it has a large component and is with $\left|S_{\ell }\right|⩽3$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|⩽6$. A proof similar to Case 1 shows that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾6n-10=20>19$, a contradiction. This implies that $\left|V\right(M\left)\right|⩽5$.

It remains to consider $10⩽f_j⩽\left|F\right|-f_i⩽19-4=15$. In this situation, $f_c⩽\left|F\right|-f_i-f_j⩽19-4-10=5$, which means that at most five faulty external edges. Since every vertex in M has exactly one external neighbor, we have $\left|V\left(M\right)\right|⩽f_c⩽5$.

Case I-3. $\left|I\right|=3$. We let $I=\{i,j,k\}$. Without loss of generality, we suppose $f_i⩽f_j⩽f_k$. Since $\left|F\right|⩽19$, we have $4⩽f_i⩽f_j⩽f_k⩽19-2\times 4=11$. If $f_j⩾8$, then $\left|F\right|=f_i+f_j+f_k⩾4+2\times 8=20$, a contradiction. Thus, $4⩽f_i⩽f_j⩽7$. We first consider $4⩽f_i⩽f_j⩽f_k⩽7=3(n-1)-5$. For each $\ell \in I$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(2), it has large components and is with $\left|S_{\ell }\right|⩽2$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|⩽6$. A proof similar to Case 1 shows that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾6n-10=20>19$, a contradiction. This implies that $\left|V\right(M\left)\right|⩽5$.

It remains to consider $8⩽f_k⩽11$. In this situation, $f_c⩽\left|F\right|-f_i-f_j-f_k⩽19-2\times 4-8=3$, which means that at most three faulty external edges. Since every vertex in M has exactly one external neighbor, we have $\left|V\left(M\right)\right|⩽f_c⩽3$.

Case I-4. $\left|I\right|=4$. We let $I=\{i,j,k,m\}$. As $f_{\ell }⩾4$ for each $\ell \in I$, we have $f_c⩽\left|F\right|-f_i-f_j-f_k-f_m⩽19-4\times 4=3$, which means that at most three faulty external edges. Since every vertex in M has exactly one external neighbor, we have $\left|V\left(M\right)\right|⩽f_c⩽3$.

For induction step, we assume $n⩾6$ and the result holds for $B{P}_{n-1}$. That is, for each $i\in I$, if $|F_i|⩽6(n-1)-11$, then $B{P}_n^i-F_i$ either is connected or contains a large component and smaller components containing totally at most five vertices. We let $S_i\subset V\left(B{P}_n^i\right)$ be the set of vertices that do not belong to the large component of $B{P}_n^i-F_i$. Obviously, if $\left|I\right|=0$, then $B{P}_n-F=B{P}_n^J-F_J$ is connected and the result holds. We consider the following cases:

Case II-1. $\left|I\right|=1$. We let $I=\left\{i\right\}$. There are two subcases depending on the range of $f_i$.

Case II-1.1. $n-1⩽f_i⩽6n-17$.

Since $B{P}_n^i$ is isomorphic to $B{P}_{n-1}$ and $f_i⩽6n-17=6(n-1)-11$, by induction hypothesis, we have $\left|S_i\right|⩽5$. Since every vertex of $B{P}_n^i$ has an external edge, there are $2^{n-1}(n-1)!$ edges between $B{P}_n^i$ and $B{P}_n^J-F_J$. Also, since $2^{n-1}(n-1)!-|S_i|⩾2^{n-1}(n-1)!-5>6n-11⩾\left|F\right|$ when $n⩾6$, the large component of $B{P}_n^i-F_i$ is connected to $B{P}_n^J-F_J$. As M is the union of smaller components of $B{P}_n-F$, this implies that $\left|V\left(M\right)\right|⩽\left|S_i\right|⩽5$.

Case II-1.2. $6n-16⩽f_i⩽6n-11$.

In this case, we have $f_c⩽\left|F\right|-f_i⩽(6n-11)-(6n-16)=5$, which means that F contains at most five faulty external edges. Since every vertex in M has exactly one external neighbor, we have $\left|V\left(M\right)\right|⩽f_c⩽5$.

Case II-2. $\left|I\right|=2$. We let $I=\{i,j\}$ and, without loss of generality, we suppose $f_i⩽f_j$. Since $\left|F\right|⩽6n-11$, we have $n-1⩽f_i⩽f_j⩽(6n-11)-(n-1)=5n-10$. If $f_i⩾4n-11$, then $f_i+f_j⩾2(4n-11)=8n-22>6n-11⩾\left|F\right|$ for $n⩾6$. Thus, it requires that $f_i⩽4n-12$. We consider the following two subcases.

Case II-2.1. $n-1⩽f_j⩽4n-11$.

In this case, we have $n-1⩽f_i⩽f_j⩽4n-11=4(n-1)-7$. For each $\ell \in I$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected; by Lemma 5(3), it contains a large component and is with $\left|S_{\ell }\right|⩽3$. Then, via a proof similar to Case 1.1, we can show that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|⩽6$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾6n-10$, a contradiction. This implies that $\left|V\right(M\left)\right|⩽5$.

Case II-2.2. $4n-10⩽f_j⩽5n-10$.

In this case, we have $n-1⩽f_i⩽min\{(6n-11)-(4n-10),4n-12\}=2n-1<4(n-1)-7$ for $n⩾6$. By Lemma 5(3), if $B{P}_n^i-F_i$ is disconnected, it contains a large component and is with $\left|S_i\right|⩽3$. For $4n-10⩽f_j⩽5n-14=5(n-1)-9$, if $B{P}_n^j-F_j$ is disconnected, by Lemma 6, it contains a large component and is with $\left|S_j\right|⩽4$. Since $2^{n-1}(n-1)!-\left|S_i\right|-\left|S_j\right|⩾2^{n-1}(n-1)!-7>6n-11⩾\left|F\right|$ when $n⩾6$, the large component of $B{P}_n^{\ell }-F_{\ell }$ for $\ell \in \{i,j\}$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|⩽7$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾6n-10$, a contradiction. Similarly, if $\left|V\right(M\left)\right|=7$, by Lemma 10, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾7n-12>6n-10$ when $n⩾6$, a contradiction. This implies that $\left|V\right(M\left)\right|⩽5$.

It remains to consider $5n-13⩽f_j⩽5n-10$. In this situation, since $\left|F\right|⩽6n-11$ and $f_i⩾n-1$, it follows that $f_c⩽\left|F\right|-f_i-f_j=(6n-11)-(n-1)-(5n-13)=3$. Thus, at most three vertices in $B{P}_n^i\cup B{P}_n^j-(F_i\cup F_j)$ cannot connect to $B{P}_n^J-F_J$ in $B{P}_n-F$, i.e., $\left|V\left(M\right)\right|⩽f_c⩽3$.

Case II-3. $\left|I\right|=3$. We let $I=\{i,j,k\}$. Without loss of generality, we suppose $f_i⩽f_j⩽f_k$. Since $\left|F\right|⩽6n-11$, we have $n-1⩽f_i⩽f_j⩽f_k⩽(6n-11)-2(n-1)=4n-9$. If $f_j⩾3n-7$, then $f_i+f_j+f_k⩾(n-1)+2(3n-7)=7n-15>6n-11⩾\left|F\right|$ for $n⩾6$. Thus, it requires that $f_i⩽f_j⩽3n-8$. We consider the following two subcases.

Case II-3.1. $n-1⩽f_i⩽f_j⩽f_k⩽3n-8=3(n-1)-5$.

For each $\ell \in I$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(2), it contains a large component and is with $\left|S_{\ell }\right|⩽2$. Then, via a proof similar to Case 1.1, we can show that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|⩽6$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾6n-10$, a contradiction. This implies that $\left|V\right(M\left)\right|⩽5$.

Case II-3.2. $n-1⩽f_i⩽f_j⩽3n-8<3n-7⩽f_k⩽4n-9$

For each $\ell \in \{i,j\}$, since $n-1⩽f_i⩽f_j⩽3n-8=3(n-1)-5$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(2), it contains a large component and is with $\left|S_{\ell }\right|⩽2$. For $3n-7⩽f_k⩽4n-11=4(n-1)-7$, if $B{P}_n^k-F_k$ is disconnected, by Lemma 5(3), it contains a large component and is with $\left|S_k\right|⩽3$. Then, via a proof similar to Case 1.1, we can show that the large component of $B{P}_n^{\ell }-F_{\ell }$ for $\ell \in \{i,j,k\}$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|⩽7$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾6n-10$, a contradiction. Similarly, if $\left|V\right(M\left)\right|=7$, by Lemma 10, $\left|F\right|⩾\left|E\right(V\left(M\right),B{P}_n-V\left(M\right)\left)\right|⩾7n-12>6n-10$ when $n⩾6$, a contradiction. This implies that $\left|V\right(M\left)\right|⩽5$.

It remains to consider $4n-10⩽f_k⩽4n-9$. In this situation, since $\left|F\right|⩽6n-11$ and $f_j⩾f_i⩾n-1$, it follows that $f_c⩽\left|F\right|-f_i-f_j-f_k⩽(6n-11)-2(n-1)-(4n-10)=1$. Thus, at most one vertex in $B{P}_n^i\cup B{P}_n^j\cup B{P}_n^k-(F_i\cup F_j\cup F_k)$ cannot connect with $B{P}_n^J-F_J$ in $B{P}_n-F$, i.e., $\left|V\left(M\right)\right|⩽f_c⩽1$.

Case II-4. $\left|I\right|=4$. We let $I=\{i,j,k,m\}$. Without loss of generality, we suppose $f_i⩽f_j⩽f_k⩽f_m$. Since $\left|F\right|⩽6n-11$, we have $n-1⩽f_i⩽f_j⩽f_k⩽f_m⩽(6n-11)-3(n-1)=3n-8$. If $f_k⩾2n-4$, then $f_i+f_j+f_k+f_m⩾2(n-1)+2(2n-4)=6n-10>6n-11⩾\left|F\right|$ for $n⩾6$. Thus, it requires that $n-1⩽f_i⩽f_j⩽f_k⩽2n-5$. We consider the following two subcases.

Case II-4.1. $n-1⩽f_i⩽f_j⩽f_k⩽f_m⩽2n-5<2(n-1)-2$.

For each $\ell \in I$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(1), it has two components, one of which is a singleton, i.e., $\left|S_{\ell }\right|=1$. A proof similar to Case 1.1 shows that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. Clearly, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|+\left|S_m\right|⩽4$.

Case II-4.2. $n-1⩽f_i⩽f_j⩽f_k⩽2n-5<2n-4⩽f_m⩽3n-8$.

For each $\ell \in \{i,j,k\}$, since $n-1⩽f_{\ell }⩽2n-5<2(n-1)-2$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(1), it contains a large component and is with $\left|S_{\ell }\right|=1$. Since $2n-4⩽f_m⩽3n-8=3(n-1)-5$, if $B{P}_n^m-F_m$ is disconnected, by Lemma 5(2), it contains a large component and is with $\left|S_m\right|⩽2$. A proof similar to Case 1.1 shows that the large component of $B{P}_n^{\ell }-F_{\ell }$ for $\ell \in I$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|+\left|S_m\right|⩽5$.

Case II-5. $\left|I\right|=5$. We let $I=\{i,j,k,m,p\}$. For each $\ell \in I$, we let $S_{\ell }\subset V\left(B{P}_n^{\ell }\right)$ be the set of vertices that do not belong to the large component of $B{P}_n^{\ell }-F_{\ell }$. Since $\left|F\right|⩽6n-11$, we have $n-1⩽f_{\ell }⩽(6n-11)-4(n-1)=2n-7<2(n-1)-2$ for $n⩾6$. Since $f_{\ell }\ne 2(n-1)-2$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(1), it has two components, one of which is a singleton, i.e., $\left|S_{\ell }\right|=1$. A proof similar to Case 1.1 shows that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. Clearly, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|+\left|S_m\right|+\left|S_p\right|⩽5$. □

4. Applications to Extra Edge Connectivity and Component Edge Connectivity

As applications of Theorem 1, we determine the relation between ${\lambda }^{\left(\ell \right)}\left(B{P}_n\right)$ and ${\lambda }_{\ell +2}\left(B{P}_n\right)$ for $3⩽\ell ⩽5$.

4.1. Relation between ${\lambda }^{\left(3\right)}\left(B{P}_n\right)$ and ${\lambda }_5\left(B{P}_n\right)$

Lemma 11.

For $n⩾5$, let $B{P}_n$ be the n-dimensional burnt pancake graph and $F\subset E\left(B{P}_n\right)$ be an arbitrary edge set. If $\left|F\right|⩽4n-4$, then $B{P}_n-F$ has at most four components.

Proof. 

Note that $\left|F\right|⩽4n-4⩽5n-9$ for $n⩾5$. By Lemma 6, if $B{P}_n-F$ is disconnected, it has a large component along with smaller components containing totally at most four vertices. Suppose that $B{P}_n-S$ has five components, four of which are singletons. By Lemma 7, isolating these four singletons requires the removal of at least $4n-3$ edges, which contradicts that $\left|F\right|⩽4n-4$. □

Theorem 2.

${\lambda }_5\left(B{P}_n\right)={\lambda }^{\left(3\right)}\left(B{P}_n\right)+3=4n-3$ for $n⩾5$.

Proof. 

Let $s=4$ and $S^{*}=arg\underset{S\subseteq V\left(B{P}_n\right)}{min}\left\{\right|E(S,G-S)|:|S|=s,$ $B{P}_n\left[S\right]$ and $B{P}_n-S$ are connected subgraphs}. As $|S^{*}|=4$ and $B{P}_n\left[S^{*}\right]$ is connected, observe from Table 1 that $B{P}_n\left[S^{*}\right]$ is a 3-path or a $K_{1,3}$. By Lemma 7, let $t=\left|E\right(S^{*},G-S^{*}\left)\right|=4n-6$ and $m=\left|E\right(B{P}_n\left[S^{*}\right]\left)\right|=3$. Let F be an edge-cut of $B{P}_n$. By Lemma 5(3), if $\left|F\right|⩽4n-7=(4n-6)-1=t-1$, then $B{P}_n-F$ has a large component along with smaller components containing totally at most $s-1=3$ vertices. This fulfills the condition of Lemma 1(i). Also, by Lemma 11, if $\left|F\right|⩽4n-4=(4n-6)+3-1=t+m-1$, then $B{P}_n-F$ has at most $s=4$ components. This fulfills the condition of Lemma 1(ii). Therefore, by Lemma 1, have ${\lambda }_{4+1}\left(B{P}_n\right)={\lambda }^{(4-1)}\left(B{P}_n\right)+m=t+m=(4n-6)+3=4n-3$ for $n⩾5$. □

4.2. Relation between ${\lambda }^{\left(4\right)}\left(B{P}_n\right)$ and ${\lambda }_6\left(B{P}_n\right)$

Lemma 12.

For $n⩾6$, let $B{P}_n$ be the n-dimensional burnt pancake graph and $F\subset E\left(B{P}_n\right)$ be an arbitrary edge set. If $\left|F\right|⩽5n-5$, then $B{P}_n-F$ has at most five components.

Proof. 

Note that $\left|F\right|⩽5n-5⩽6n-11$ for $n⩾6$. By Theorem 1, if $B{P}_n-F$ is disconnected, it has a large component and smaller components containing totally at most five vertices. Suppose that $B{P}_n-S$ has six components, five of which are singletons. By Lemma 8, isolating these five singletons requires the removal of at least $5n-4$ edges, which contradicts that $\left|F\right|⩽5n-5$. □

Theorem 3.

${\lambda }_6\left(B{P}_n\right)={\lambda }^{\left(4\right)}\left(B{P}_n\right)+4=5n-4$ for $n⩾6$.

Proof. 

Let $s=5$ and $S^{*}=arg\underset{S\subseteq V\left(B{P}_n\right)}{min}\left\{\right|E(S,G-S)|:|S|=s,$ $B{P}_n\left[S\right]$ and $B{P}_n-S$ are connected subgraphs}. As $|S^{*}|=5$ and $B{P}_n\left[S^{*}\right]$ is connected, observe from Table 2 that $B{P}_n\left[S^{*}\right]$ is a 4-path or a tree with 5 vertices (including $K_{1,4}$). By Lemma 8, let $t=\left|E\right(S^{*},G-S^{*}\left)\right|=5n-8$ and $m=\left|E\right(B{P}_n\left[S^{*}\right]\left)\right|=4$. Let F be an edge-cut of $B{P}_n$. By Lemma 6, if $\left|F\right|⩽5n-9=(5n-8)-1=t-1$, then $B{P}_n-F$ has a large component and smaller components containing totally at most $s-1=4$ vertices. This fulfills the condition of Lemma 1(i). Also, by Lemma 12, if $\left|F\right|⩽5n-5=(5n-8)+4-1=t+m-1$, then $B{P}_n-F$ has at most $s=5$ components. This fulfills the condition of Lemma 1(ii). Therefore, by Lemma 1, have ${\lambda }_{5+1}\left(B{P}_n\right)={\lambda }^{(5-1)}\left(B{P}_n\right)+m=t+m=(5n-8)+4=5n-4$ for $n⩾6$. □

4.3. Relation between ${\lambda }^{\left(5\right)}\left(B{P}_n\right)$ and ${\lambda }_7\left(B{P}_n\right)$

Lemma 13.

For $n⩾6$, let $B{P}_n$ be the n-dimensional burnt pancake graph and $F\subset E\left(B{P}_n\right)$ be an arbitrary edge set. If $\left|F\right|⩽6n-6$, then $B{P}_n-F$ has at most six components.

Proof. 

Let M be the union of smaller components of $B{P}_n-F$ and let $c\left(M\right)$ be the such number of components in M. By the definition of M, it suffices to show that $c\left(M\right)⩽5$. Since $\left|F\right|⩽6n-6$ and $f_i⩾n-1$ for $i\in I$, have $\left|I\right|⩽6$. Then, $\left|J\right|=2n-\left|I\right|⩾2n-6⩾5$ when $n⩾6$. With reasoning similar to Remark 1 in the proof of Theorem 1, it is shown that $B{P}_n^j-Fj$ is connected for each $j\in J$ and the following remark is further obtained.

Remark 2.

$B{P}_n^J-F_J$ is connected.

Obviously, if $\left|I\right|=0$, then $B{P}_n-F=B{P}_n^J-F_J$ is connected and the result holds. Now consider $1⩽\left|I\right|⩽6$. For each $i\in I$, let $S_i\subset V\left(B{P}_n^i\right)$ be the set of vertices that do not belong to the large component of $B{P}_n^i-F_i$.

Case 1. $\left|I\right|=1$. Let $I=\left\{i\right\}$. There are two subcases depending on the range of $f_i$.

Case 1.1. $n-1⩽f_i⩽6n-17=6(n-1)-11$.

Since $B{P}_n^i$ is isomorphic to $B{P}_{n-1}$, by Theorem 1, $B{P}_n^i-F_i$ has a large component and is with $\left|S_i\right|⩽5$. As every vertex of $B{P}_n^i$ has an external edge, there are $2^{n-1}(n-1)!$ edges between $B{P}_n^i$ and $B{P}_n^J-F_J$. Also, since $2^{n-1}(n-1)!-5>6n-6⩾\left|F\right|$ when $n⩾6$, the large component of $B{P}_n^i-F_i$ is connected to $B{P}_n^J-F_J$. This implies that $c\left(M\right)⩽\left|V\left(M\right)\right|⩽\left|S_i\right|⩽5$.

Case 1.2. $6n-16⩽f_i⩽6n-6$.

In this case, there is $f_c⩽\left|F\right|-f_i⩽(6n-6)-(6n-16)=10$, which means that F contains at most ten faulty external edges. Since every vertex in M has exactly one external neighbor, there is $\left|V\left(M\right)\right|⩽f_c⩽10$. If $\left|V\right(M\left)\right|⩽5$, it is clear that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾|N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|⩾6n-5>6n-6$, a contradiction. Similarly, if $\left|V\right(M\left)\right|=7$, by Lemma 10, $\left|F\right|⩾|N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|⩾7n-6>6n-6$, a contradiction. Now deal with the situations for $8⩽\left|V\right(M\left)\right|⩽10$ as follows.

Case 1.2.1. $\left|V\right(M\left)\right|=8$. Let $V\left(M\right)=V\left(M_1\right)\cup \left\{x\right\}$, where $\left|V\right(M_1\left)\right|=7$. By Lemma 10, $|N_{E\left(B{P}_n\right)}\left(V\left(M_1\right)\right)|⩾7n-6$. Clearly, $|N_{E\left(B{P}_n\right)}\left(x\right)|=n$ and x may connect to at most 7 vertices in $M_1$, i.e., $\left|E\right(\left\{x\right\},M_1\left)\right|⩽7$. Thus,

$$\begin{array}{ccc}\hfill |N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|& =& |N_{E\left(B{P}_n\right)}\left(V\left(M_1\right)\right)|+|N_{E\left(B{P}_n\right)}\left(x\right)|-|E(\left\{x\right\},M_1)|\hfill \\ & ⩾& (7n-6)+n-7=8n-13>6n-6⩾\left|F\right|\hfill \end{array}$$

when $n⩾6$, a contradiction.

Case 1.2.2. $\left|V\right(M\left)\right|=9$. Let $V\left(M\right)=V\left(M_1\right)\cup \left\{x\right\}$, where $\left|V\right(M_1\left)\right|=8$. By Case 1.2.1, $|N_{E\left(B{P}_n\right)}\left(V\left(M_1\right)\right)|⩾8n-13$. Clearly, $|N_{E\left(B{P}_n\right)}\left(x\right)|=n$ and x may connect to at most 8 vertices in $M_1$, i.e., $\left|E\right(\left\{x\right\},M_1\left)\right|⩽8$. Thus,

$$\begin{array}{ccc}\hfill |N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|& =& |N_{E\left(B{P}_n\right)}\left(V\left(M_1\right)\right)|+|N_{E\left(B{P}_n\right)}\left(x\right)|-|E(\left\{x\right\},M_1)|\hfill \\ & ⩾& (8n-13)+n-8=9n-21>6n-6⩾\left|F\right|\hfill \end{array}$$

when $n⩾6$, a contradiction.

Case 1.2.3. $\left|V\right(M\left)\right|=10$. Let $V\left(M\right)=V\left(M_1\right)\cup \{x,y\}$, where $\left|V\right(M_1\left)\right|=8$. By Case 1.2.1, $|N_{E\left(B{P}_n\right)}\left(V\left(M_1\right)\right)|⩾8n-13$. First, consider $xy$ forms an edge in M. Then, $|N_{E\left(B{P}_n\right)}\left(\{x,y\}\right)|=2n-1$ and x (resp., y) may connect to at most eight vertices or $n-1$ vertices (if $6⩽n⩽8$) in $M_1$. That is, $\left|E\right(\left\{x\right\},M_1\left)\right|⩽min\{8,n-1\}$ and $\left|E\right(\left\{y\right\},M_1\left)\right|⩽min\{8,n-1\}$. Since $xy\in E\left(B{P}_n\right)$ and the girth of $B{P}_n$ is 8, x and y cannot be adjacent to a vertex in $M_1$ simultaneously. Thus, $\left|E\right(\{x,y\},M_1\left)\right|⩽min\{8,n-1\}⩽8$ and

$$\begin{array}{ccc}\hfill |N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|& =& |N_{E\left(B{P}_n\right)}\left(V\left(M_1\right)\right)|+|N_{E\left(B{P}_n\right)}\left(\{x,y\}\right)|-|E(\{x,y\},M_1)|\hfill \\ & ⩾& (8n-13)+(2n-1)-8=10n-22>6n-6⩾\left|F\right|\hfill \end{array}$$

when $n⩾6$, a contradiction. Next, suppose x and y are singletons in M. Then, $|N_{E\left(B{P}_n\right)}$$\left(\right\{x,y\left\}\right)|=2n$ and $|E(\{x,y\},M_1)|⩽min\{16,2(n-1)\}⩽16$. Thus,

$$\begin{array}{ccc}\hfill |N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|& =& |N_{E\left(B{P}_n\right)}\left(V\left(M_1\right)\right)|+|N_{E\left(B{P}_n\right)}\left(\{x,y\}\right)|-|E(\{x,y\},M_1)|\hfill \\ & ⩾& (8n-13)+2n-16=10n-29>6n-6⩾\left|F\right|\hfill \end{array}$$

when $n⩾6$, a contradiction.

Based on the discussion of the above situations, conclude $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

Case 2. $\left|I\right|=2$. Let $I=\{i,j\}$ and, without loss of generality, suppose $f_i⩽f_j$. Since $\left|F\right|⩽6n-6$, there is $n-1⩽f_i⩽f_j⩽6n-6-(n-1)=5n-5$. Consider the following three subcases.

Case 2.1. $n-1⩽f_i⩽f_j⩽4n-11$.

In this case, there is $n-1⩽f_i⩽f_j⩽4n-11=4(n-1)-7$. For each $\ell \in I$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(3), it contains a large component and is with $|S_{\ell }|⩽3$. Then, via a proof similar to Case 1.1, it can be shown that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|⩽6$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾|N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|=6n-5$, a contradiction. This implies that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

Case 2.2. $n-1⩽f_i⩽4n-11<4n-10⩽f_j⩽5n-5$.

In this case, there is $n-1⩽f_i⩽4n-11=4(n-1)-7$. By Lemma 5(3), if $B{P}_n^i-F_i$ is disconnected, it contains a large component and is with $\left|S_i\right|⩽3$. For $4n-10⩽f_j⩽5n-14=5(n-1)-9$, if $B{P}_n^j-F_j$ is disconnected, by Lemma 6, it contains a large component and is with $\left|S_i\right|⩽4$. Then, via a proof similar to Case 1.1, it can be shown that the large component of $B{P}_n^{\ell }-F_{\ell }$ for $\ell \in I$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|⩽7$. If $6⩽\left|V\right(M\left)\right|⩽7$, a contradiction can be acquired through an argument similar to Case 1.2. This implies that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

It remains to consider $5n-13⩽f_j⩽5n-5$. In this situation, since $\left|F\right|⩽6n-6$, there is $f_c⩽\left|F\right|-f_i-f_j⩽(6n-6)-(n-1)-(5n-13)=8$. Thus, at most eight vertices in $B{P}_n^I-F_I$ cannot connect to $B{P}_n^J-F_J$ in $B{P}_n-F$, i.e., $\left|V\left(M\right)\right|⩽f_c⩽8$. If $6⩽\left|V\right(M\left)\right|⩽8$,a contradiction can be acquired through an argument similar to Case 1.2. Thus, $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

Case 2.3. $4n-10⩽f_i⩽f_j⩽5n-5$.

In this case, $6n-6⩾f_i+f_j⩾2(4n-10)=8n-20$, which leads to $6⩽n⩽7$. Note that $f_c⩽\left|F\right|-f_i-f_j⩽(6n-6)-2(4n-10)=14-2n$. Thus, $0⩽f_c⩽2$ and at most two vertices in $B{P}_n^I-F_I$ cannot connect with $B{P}_n^J-F_J$ in $B{P}_n-F$, i.e., $\left|V\left(M\right)\right|⩽f_c⩽2$. It is clear that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽2$.

Case 3. $\left|I\right|=3$. Let $I=\{i,j,k\}$. Without loss of generality, suppose $f_i⩽f_j⩽f_k$. Since $\left|F\right|⩽6n-6$, there is $n-1⩽f_i⩽f_j⩽f_k⩽(6n-6)-2(n-1)=4n-4$. If $f_i⩾3n-7$, then $f_i+f_j+f_k⩾3(3n-7)=9n-21>6n-6⩾\left|F\right|$ for $n⩾6$. Thus, it requires that $n-1⩽f_i⩽3n-8$. Consider the following three subcases.

Case 3.1. $n-1⩽f_i⩽f_j⩽f_k⩽3n-8=3(n-1)-5$.

For each $\ell \in I$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(2), it contains a large component and is with $\left|S_{\ell }\right|⩽2$. Then, via a proof similar to Case 1.1, it can be shown that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|⩽6$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾|N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|⩾6n-5>6n-6$, a contradiction. This implies that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

Case 3.2. $n-1⩽f_i⩽f_j⩽3n-8<3n-7⩽f_k⩽4n-4$.

For each $\ell \in \{i,j\}$, since $n-1⩽f_i⩽f_j⩽3n-8=3(n-1)-5$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(2), it contains a large component and is with $\left|S_{\ell }\right|⩽2$. For $3n-7⩽f_k⩽4n-11=4(n-1)-7$, if $B{P}_n^k-F_k$ is disconnected, by Lemma 5(3), it contains a large component and is with $\left|S_k\right|⩽3$. Then, via a proof similar to Case 1.1, it can b shown that the large component of $B{P}_n^{\ell }-F_{\ell }$ for $\ell \in I$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|⩽7$. If $6⩽\left|V\right(M\left)\right|⩽7$, a contradiction can be acquired through an argument similar to Case 1.2. Thus, $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

It remains to consider $4n-10⩽f_k⩽4n-4$. In this situation, since $\left|F\right|⩽6n-6$ and $f_j⩾f_i⩾n-1$, there is $f_c⩽\left|F\right|-f_i-f_j-f_k⩽(6n-6)-2(n-1)-(4n-10)=6$. Thus, at most six vertices in $B{P}_n^I-F_I$ cannot connect with $B{P}_n^J-F_J$ in $B{P}_n-F$, i.e., $\left|V\left(M\right)\right|⩽f_c⩽6$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾|N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|⩾6n-5>6n-6$, a contradiction. This implies that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

Case 3.3. $n-1⩽f_i⩽3n-8<3n-7⩽f_j⩽f_k⩽4n-4$.

In this case, $6n-6⩾f_i+f_j+f_k⩾(n-1)+2(3n-7)=7n-15$, which leads to $6⩽n⩽9$. Note that $f_c⩽\left|F\right|-f_i-f_j-f_k⩽(6n-6)-(n-1)-2(3n-7)=9-n$. Thus, $0⩽f_c⩽3$ and at most three vertices in $B{P}_n^I-F_I$ cannot connect with $B{P}_n^J-F_J$ in $B{P}_n-F$, i.e., $\left|V\left(M\right)\right|⩽f_c⩽3$. It is clear that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽3$.

Case 4. $\left|I\right|=4$. Let $I=\{i,j,k,m\}$. Without loss of generality, suppose $f_i⩽f_j⩽f_k⩽f_m$. Since $\left|F\right|⩽6n-6$, there is $n-1⩽f_i⩽f_j⩽f_k⩽f_m⩽(6n-6)-3(n-1)=3n-3$. If $f_i⩾2n-4$, then $f_i+f_j+f_k+f_m⩾4(2n-4)=8n-16>6n-6⩾\left|F\right|$ for $n⩾6$. Thus, it requires that $n-1⩽f_i⩽2n-5$. Also, if $f_k⩾3n-7$, then $f_i+f_j+f_k+f_m⩾2(n-1)+2(3n-7)=8n-16>6n-6⩾\left|F\right|$ for $n⩾6$. Thus, it requires that $n-1⩽f_j⩽f_k⩽3n-8$. Consider the following two subcases.

Case 4.1. $n-1⩽f_i⩽2n-5$, $n-1⩽f_j⩽f_k⩽f_m⩽3n-8$.

In this case, there is $n-1⩽f_i⩽2n-5<2(n-1)-2$. If $B{P}_n^i-F_i$ is disconnected, by Lemma 5(1), it has two components, one of which is a singleton, i.e., $\left|S_i\right|=1$. For $\ell \in \{j,k,m\}$, since $n-1⩽f_{\ell }⩽3n-8<3(n-1)-5$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(2), it contains a large component and is with $\left|S_k\right|⩽2$. A proof similar to Case 1.1 shows that the large component of $B{P}_n^{\ell }-F_{\ell }$ for $\ell \in I$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\left(M\right)\right|⩽\left|S_i\right|+\left|S_j\right|+\left|S_k\right|+\left|S_m\right|⩽7$. If $6⩽\left|V\right(M\left)\right|⩽7$, a contradiction can be acquired through an argument similar to Case 1.2. Thus, $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

Case 4.2. $n-1⩽f_i⩽f_j⩽f_k⩽3n-8<3n-7⩽f_m⩽3n-3$.

In this case, $f_c⩽\left|F\right|-f_i-f_j-f_k-f_m⩽(6n-6)-3(n-1)-(3n-7)=2$. Thus, at most two vertices in $B{P}_n^I-F_I$ cannot connect with $B{P}_n^J-F_J$ in $B{P}_n-F$, i.e., $\left|V\left(M\right)\right|⩽f_c⩽2$. It is clear that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽2$.

Case 5. $\left|I\right|=5$. Let $I=\{i,j,k,m,p\}$. Without loss of generality, suppose $f_i⩽f_j⩽f_k⩽f_m⩽f_p$. Since $\left|F\right|⩽6n-6$, there is $n-1⩽f_i⩽f_j⩽f_k⩽f_m⩽f_p⩽(6n-6)-4(n-1)=2n-2$. If $f_m⩾2n-4$, then $f_i+f_j+f_k+f_m+f_p⩾3(n-1)+2(2n-4)=7n-11>6n-6⩾\left|F\right|$ for $n⩾6$. Thus, it requires that $n-1⩽f_i⩽f_j⩽f_k⩽f_m⩽2n-5$. Consider the following two subcases.

Case 5.1. $n-1⩽f_i⩽f_j⩽f_k⩽f_m⩽f_p⩽2n-5$.

For each $\ell \in I$, if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, by Lemma 5(1), $B{P}_n^{\ell }-F_{\ell }$ has two components, one of which is a singleton, i.e., $\left|S_i\right|=1$. A proof similar to Case 1.1 shows that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\right(M\left)\right|⩽5$. This leads to $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$.

Case 5.2. $n-1⩽f_i⩽f_j⩽f_k⩽f_m⩽2n-5<2n-4⩽f_p⩽2n-2$.

In this case, $f_c⩽\left|F\right|-f_i-f_j-f_k-f_m⩽(6n-6)-4(n-1)-(2n-4)=2$. Thus, at most two vertices in $B{P}_n^I-F_I$ cannot connect with $B{P}_n^J-F_J$ in $B{P}_n-F$, i.e., $\left|V\left(M\right)\right|⩽f_c⩽2$. It is clear that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽2$.

Case 6. $\left|I\right|=6$. Let $I=\{i,j,k,m,p,q\}$. Since $\left|F\right|⩽6n-6$, there is $f_i=f_j=f_k=f_m=f_p=f_q=n-1$ and $f_c=0$. Since $f_{\ell }=n-1<2(n-1)-2$ for $n⩾6$, by Lemma 5(1), if $B{P}_n^{\ell }-F_{\ell }$ is disconnected, then $B{P}_n^{\ell }-F_{\ell }$ has two components, one of which is a singleton, i.e., $\left|S_{\ell }\right|=1$. As $n⩾6=\left|I\right|$, by Lemma 2(1), there exists ${\ell }^{\prime }\in \langle n\rangle \setminus (I\cup \left\{\overline{\ell }\right\})$ such that $|E_{\ell ,{\ell }^{\prime }}\left(B{P}_n\right)|-(f_{\ell }+1)=(n-2)!\times 2^{n-2}-(n-1)-1>0$. This implies that the large component of $B{P}_n^{\ell }-F_{\ell }$ is connected to $B{P}_n^J-F_J$. Thus, $\left|V\right(M\left)\right|⩽6$. If $\left|V\right(M\left)\right|=6$, by Lemma 9, $\left|F\right|⩾|N_{E\left(B{P}_n\right)}\left(V\left(M\right)\right)|⩾6n-5>6n-6$, a contradiction. This implies that $c\left(M\right)⩽\left|V\right(M\left)\right|⩽5$. □

Theorem 4.

${\lambda }_7\left(B{P}_n\right)={\lambda }^{\left(5\right)}\left(B{P}_n\right)+5=6n-5$ for $n⩾6$.

Proof. 

Let $s=6$ and $S^{*}=arg\underset{S\subseteq V\left(B{P}_n\right)}{min}\left\{\right|E(S,G-S)|:|S|=s,$ $B{P}_n\left[S\right]$ and $B{P}_n-S$ are connected subgraphs}. As $|S^{*}|=6$ and $B{P}_n\left[S^{*}\right]$ is connected, it can be observed from Table 3 that $B{P}_n\left[S^{*}\right]$ is a 5-path or a tree with 6 vertices (including $K_{1,5}$). By Lemma 9, let $t=\left|E\right(S^{*},G-S^{*}\left)\right|=6n-10$ and $m=\left|E\right(B{P}_n\left[S^{*}\right]\left)\right|=5$. Let F be an edge-cut of $B{P}_n$. By Theorem 1, if $\left|F\right|⩽6n-11=(6n-10)-1=t-1$, then $B{P}_n-F$ has a large component along with smaller components containing totally at most $s-1=5$ vertices. This fulfills the condition of Lemma 1(i). Also, by Lemma 13, if $\left|F\right|⩽6n-6=(6n-10)+5-1=t+m-1$, then $B{P}_n-F$ has at most $s=6$ components. This fulfills the condition of Lemma 1(ii). Therefore, by Lemma 1, there is ${\lambda }_{6+1}\left(B{P}_n\right)={\lambda }^{(6-1)}\left(B{P}_n\right)+m=t+m=(6n-10)+5=6n-5$ for $n⩾6$. □

5. Concluding Remarks

For burnt pancake graph $B{P}_n$, this paper shows that when removing any edge subset with a size of approximately six times $\lambda \left(B{P}_n\right)$, the surviving graph possesses the “linearly many faults" property. Applying this property, we attain ${\lambda }^{\left(h\right)}\left(B{P}_n\right)$ and ${\lambda }_r\left(B{P}_n\right)$. Specifically, we prove that ${\lambda }_5\left(B{P}_n\right)={\lambda }^{\left(3\right)}\left(B{P}_n\right)+3=4n-3$ for $n⩾5$; ${\lambda }_6\left(B{P}_n\right)={\lambda }^{\left(4\right)}\left(B{P}_n\right)+4=5n-4$ and ${\lambda }_7\left(B{P}_n\right)={\lambda }^{\left(5\right)}\left(B{P}_n\right)+5=6n-5$ for $n⩾6$, as summarized in

Table 4. The comparison of ${\lambda }^{\left(h\right)}\left(B{P}_n\right)$ and ${\lambda }_r\left(B{P}_n\right)$.

${\mathit{\lambda }}^{\left(\mathit{h}\right)}\left({\mathit{BP}}_{\mathit{n}}\right)$	Ref.	${\mathit{\lambda }}_{\mathit{r}}\left({\mathit{BP}}_{\mathit{n}}\right)$	Ref.
${\lambda }^{\left(1\right)}\left(B{P}_n\right)=2n-2$	[30]	${\lambda }_3\left(B{P}_n\right)=2n-1$	[30]
${\lambda }^{\left(2\right)}\left(B{P}_n\right)=3n-4$		${\lambda }_4\left(B{P}_n\right)=3n-2$
${\lambda }^{\left(3\right)}\left(B{P}_n\right)=4n-6$	Theorem 2	${\lambda }_5\left(B{P}_n\right)=4n-3$	Theorem 2
${\lambda }^{\left(4\right)}\left(B{P}_n\right)=5n-8$	Theorem 3	${\lambda }_6\left(B{P}_n\right)=5n-4$	Theorem 3
${\lambda }^{\left(5\right)}\left(B{P}_n\right)=6n-10$	Theorem 4	${\lambda }_7\left(B{P}_n\right)=6n-5$	Theorem 4

.

For ℓ-componen edge connectivity and h-extra edge connectivity with higher ℓ and h, e.g., $h=6$ and $\ell =8$, since we showed in Lemma 10 that $E(W,B{P}_n-W)|⩾7n-12$ and $|N_{E\left(B{P}_n\right)}\left(W\right)|⩾7n-6$ for $W\subset V\left(B{P}_n\right)$ with $\left|W\right|=7$, this prompts us to have the following conjecture:

Conjecture 1.

${\lambda }_8\left(B{P}_n\right)={\lambda }^{\left(6\right)}\left(B{P}_n\right)+6=7n-6$ for $n⩾6$.

Obviously, to affirm the above conjecture is equivalent to showing that the following two implications hold for $n⩾7$ and any edge set $F\subset E(B{P}_n$): (i) If $\left|F\right|⩽7n-13$, then $B{P}_n-F$ either is connected or contains a large component along with smaller components containing totally at most six vertices. (ii) if $\left|F\right|⩽7n-7$, then $B{P}_n-F$ has at most seven components.

Similarly, as $B{P}_n$ is n-regular and its girth is eight, we are easy to check that $\left|E\right(C_8,B{P}_n-C_8\left)\right|=8n-16$ and $|N_{E\left(B{P}_n\right)}\left(C_8\right)|=8n-8$. Based on the relationship of ${\lambda }^{\left(h\right)}\left(G\right)$ and ${\lambda }_r\left(G\right)$ for a regular graph G [30], we also have the following conjecture:

Conjecture 2.

${\lambda }_9\left(B{P}_n\right)={\lambda }^{\left(7\right)}\left(B{P}_n\right)+8=8n-8$ for $n⩾6$.

To prove this conjecture, we need to show that when removing any edge subset with a size approximately of eight times $\lambda \left(B{P}_n\right)$, the surviving graph still retains the “linearly many faults” property. With the increase in the removal of edges, the situation becomes more complex, and it is an interesting and challenging research topic.

We conclude this paper by discussing some of its limitations against real-world instances. Even though various interconnection networks have specific structural phenomena when a linear number of vertices or edges fail, do these phenomena occur frequently? Since most research considers vertex or edge failures in a network to be random and uncorrelated, it ignores possible events that cause components close to each other to fail simultaneously with a higher probability. In this case, is there a more reasonable evaluation measure combining h-extra edge connectivity or ℓ-component edge connectivity that can genuinely reflect this phenomenon?