Two-Disjoint-Cycle-Cover Pancyclicity of Dragonfly Networks

Abstract

Interconnection networks (often modeled as graphs) are critical for high-performance computing systems, as they have significant impact on performance metrics like latency and bandwidth. The dragonfly network, denoted as $D(n,r)$, is a promising topology owing to its modularity, low diameter, and cost-effectiveness. Ensuring reliability and efficiency in these networks requires robust cycle embedding properties. The two-disjoint-cycle-cover pancyclicity ensures that the network can be partitioned into two vertex-disjoint cycles of any feasible length. This suggests potential advantages for improving fault tolerance and load balancing strategies in interconnection networks. Formally, a graph G is called two-disjoint-cycle-cover $[a_1,a_2]$-pancyclic if for any integer ℓ satisfying $a_1\le 𝓁\le a_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in G such that $|V(C_1)|=𝓁$ and $|V(C_2)|=|V(G)|-𝓁$. While prior work has established Hamiltonicity and pancyclicity for $D(n,r)$, the two-disjoint-cycle-cover problem remains unexplored. This paper fills this gap by proving that $D(n,r)$ is two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋]$-pancyclic with $n\ge 3$ and $r\ge 2$, generalizing existing knowledge. Moreover, it can be obtained that $D(n,r)$ is vertex-disjoint-cycle-coverable. Our proof employs a constructive method with case analysis, ensuring the existence of such cycles.

1. Introduction

Interconnection networks are critical for high-performance computing (HPC) systems because they directly impact performance metrics such as latency, bandwidth, and scalability. Among various network topologies, the dragonfly network $D(n,r)$—introduced by Kim et al. [1]—is a symmetric graph where vertices are partitioned into $g=nr+1$ groups. Each group is a complete graph on n vertices, and each vertex has r external edges that connect to other groups following a predefined rule. Figure 1 provides an example of $D(n,r)$ for $n=4, r=2, g=9$. The dragonfly network topology enables efficient data exchange and has gained attention due to its modularity, low diameter, and cost-effectiveness, making it a promising candidate for large-scale HPC applications.

Graph theory provides a foundation for studying interconnection networks, where cycle embedding properties are essential for assessing structural robustness and reliability. Given a graph $G=(V(G),E(G))$, $V(G)$ and $E(G)$ denotes all the vertices and edges in G, respectively. The number of elements in $V(G)$, written as $|V(G)|$, is known as the order of G. Given a vertex $x\in V(G)$, let $N_G(x)$ denote the neighborhood of x in G. A complete graph having n vertices, indicated by $K_n$, is such that each vertex is adjacent to every other vertex within the graph. For a cycle C (path P), the length of a cycle (path), denoted by $|V(C)|$ ($|V(P)|$), refers to the number of vertices within the cycle (path). A cycle in graph G that includes every single vertex of G is known as a Hamiltonian cycle. A graph that possesses a Hamiltonian cycle is called Hamiltonian. A graph with an order of n is regarded to be pancyclic if it has cycles of all possible lengths ranging from 3 to n. In particular, we term G as vertex-pancyclic (respectively, edge-pancyclic) when every vertex (respectively, every edge) is included in a cycle whose length ℓ satisfies $3\le 𝓁\le n$. In graph G, when there is a set of ℓ cycles $C_1,C_2,\dots ,C_{𝓁}$ whose vertex sets are disjoint and $V(G)={\cup }_{i=1}^{𝓁}V(C_i)$, we say the set covers G and is called an ℓ-disjoint-cycle-cover. Prior research has established fundamental properties of $D(n,r)$, such as Hamiltonian and pancyclicity [2].

A significant extension in graph theory is the concept of two-disjoint-cycle-cover pancyclicity introduced by Kung and Chen [3], which integrates the concepts of disjoint-cycle-cover and pancyclicity and ensures that a graph can be partitioned into two vertex-disjoint cycles of variable lengths. Formally, a graph G is two-disjoint-cycle-cover $[a_1,a_2]$-pancyclic if for any integer ℓ with $a_1\le 𝓁\le a_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in G such that $|V(C_1)|=𝓁$ and $|V(C_2)|=|V(G)|-𝓁$. This property has been extensively investigated in other network topologies (e.g., crossed cubes [3], locally twisted cubes [4], bipartite generalized hypercube [5], bipartite hypercube-like networks [6], balanced hypercubes [7], augmented cubes [8,9], alternating group graph [10], data center networks [11], bubble-sort star graphs [12], split-star networks [13], and star graphs [14]), as it enhances fault-tolerant routing and load balancing capabilities. However, for dragonfly networks, the two-disjoint-cycle-cover pancyclicity problem remains unexplored.

This paper fills the above theoretical gap by proving that $D(n,r)$ is two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋]$-pancyclic, where $n\ge 3$ and $r\ge 2$. In other words, we can find two cycles $C_1$ and $C_2$ in $D(n,r)$ with disjoint vertex sets, where $|V(C_1)|=𝓁$ and $|V(C_2)|=|V(D(n,r))|-𝓁$ with $3\le 𝓁\le ⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋$. Our result expands the existing knowledge in [2] regarding the Hamiltonicity and pancyclicity of dragonfly networks. Moreover, we can obtain $D(n,r)$ is vertex-disjoint-cycle-coverable.

The rest of the paper is organized as follows: Section 2 presents the definitions and preliminary knowledge that will be employed across the whole paper. Section 3 and Section 4 prove the main theorem for $D(4,r)$ and $D(n,r)$ with $n\ge 5$, respectively. Finally, Section 5 concludes the paper and provides some perspectives for future work.

2. Preliminaries

The dragonfly network $D(n,r)$ is defined as follows.

Definition 1

([15]). Let $n,r,g$ be any positive integers that satisfy the condition $g=nr+1$. The definition of $D(n,r)$ is presented in the following manner.

1.

The vertex set of $D(n,r)$ is given by $V(D(n,r))={\cup }_{i=0}^{g-1}V(G_i)$, where for each i from 0 to $g-1$, $G_i=(V_i,E_i)$ is a complete subgraph isomorphic to $K_n$, representing the i-th group within $D(n,r)$. Moreover, $V(G_i)$ and $V(G_j)$ are disjoint, i.e., $V(G_i)\cap V(G_j)=\varnothing$, where $0\le i, j\le g-1$ and $i\ne j$;

2.

Any vertex of $D(n,r)$ can be labeled by $(x,y)$, which denotes vertex y belonging to group x, with $0\le x\le g-1$ and $0\le y\le n-1$;

3.

The vertex $u=(x_1,y_1)$ is adjacent to vertex $v=(x_2,y_2)$ with $x_1\ne x_2$ if and only if $y_2=n-1-y_1$, $x_2=(r{y}_1+x_1+k) \mathrm{mod} g$, where k ranges from 1 to r.

Obviously, $D(n,r)$ is a graph with a regularity of $(n+r-1)$ and contains $ng$ vertices. For any $u\in V(G_i)$, we call $V(G_i)-\left\{u\right\}$ the internal neighbor set of u. In contrast, the external neighbor set refers to the set of r vertices from other groups adjacent to u. Similarly, for any edge $e\in E(D(n,r))$, we call e an internal edge if $e\in E(G_i)$; elsewise, e is called an external edge.

There are some results in [2] about the Hamiltonicity and pancyclicity of dragonfly networks as follows:

Theorem 1

([2]). For $n\ge 1$ and $r\ge 2$, $D(n,r)$ is Hamiltonian.

Theorem 2

([2]). For $n\ge 4$ and $r\ge 2$, $D(n,r)$ is vertex-pancyclic.

By Definition 1, precisely one external edge exists between any two different groups. The endpoints of this edge are determined by the following theorem [15], which provides a technical tool that is used in the proof of our main result.

Theorem 3

([15]). The external edge connecting group i and group j is $((i,\lfloor {\displaystyle \frac{j-i-1}{r}}\rfloor ),(j,n-1-\lfloor {\displaystyle \frac{j-i-1}{r}}\rfloor ))$, with $i<j$.

Lemma 1

([2]). For $0\le i\le g-1$, $0\le y\le n-1$ and $1\le l\le r-1$, we have $((yr+i+d) \mathrm{mod} g,n-y-1)\in N((i,y))\cap N((i+l,y))$, where $l+1\le d\le r$.

Lemma 2.

For $n\ge 4, r\ge 2$, a Hamiltonian cycle can be found in $V(D(n,r))-{\cup }_{j=0}^{\alpha }V(G_j)$ where $\alpha =0,1,2\cdots ,(n-1)r-2$.

Proof of Lemma 2.

For $\alpha \in \{0,1,\cdots ,(n-1)r-2\}$, by Theorem 3, there is an external edge $e=((\alpha +1,\lfloor {\displaystyle \frac{g-\alpha -3}{r}}\rfloor ), (g-1,n-1-\lfloor {\displaystyle \frac{g-\alpha -3}{r}}\rfloor )$ connecting group $G_{\alpha +1}$ and group $G_{g-1}$. Note that $\lfloor {\displaystyle \frac{g-\alpha -3}{r}}\rfloor \ne 0$; otherwise $g-\alpha -3<r$, which implies $\alpha >g-r-3=(nr+1)-r-3=(n-1)r-2$, a contradiction. Since $G_i\cong K_n$ with $0\le i\le g-1$, between any pair of vertices in $V(G_i)$, we can find a Hamiltonian path. Let $P_{\alpha +1}$ be a Hamiltonian path between $(\alpha +1,0)$ and $(\alpha +1,\lfloor {\displaystyle \frac{g-\alpha -3}{r}}\rfloor )$ in $G_{\alpha +1}$, $P_{g-1}$ be a Hamiltonian path between $(g-1,n-1)$ and $(g-1,n-1-\lfloor {\displaystyle \frac{g-\alpha -3}{r}}\rfloor )$ in $G_{g-1}$, $P_a$ be a Hamiltonian path between $(a,n-1)$ and $(a,0)$ in $G_a$ for any $a=\alpha +2,\alpha +3,\cdots g-2$.

By Definition 1, $((i,0),(i+1,n-1))$ is an edge in $D(n,r)$ with $0\le i\le g-1$. Thus, in $V(D(n,r))-{\cup }_{j=0}^{\alpha }V(G_j)$, we can find a Hamiltonian cycle:

$$C=\bigcup _{a=\alpha +1}^{g-1}{P}_a\cup (\bigcup _{i=\alpha +1}^{g-2}((i,0),(i+1,n-1)))\cup \left\{e\right\}.$$

□

Since $G_i$ is a complete graph for $i=0,1,\cdots ,g-1$, there exists a Hamiltonian path $P_i$ between vertices $(i,0)$ and $(i,n-1)$. By Definition 1, $((i,0),(i+1,n-1))\in E(D(n,r))$. By Theorem 1, there is a Hamiltonian cycle (see Figure 2)

$$C_0=\bigcup _{j=0}^{g-1}{P}_j\cup (\bigcup _{i=1}^{g-1}((i,0),(i+1,n-1))).$$

The cycle $C_0$ will play a crucial role in the proof of our result.

Let us consider the two-disjoint-cycle-cover pancyclicity problem of $D(n,r)$.

Some Individual Cases:

If $n=1$, by Definition 1, $D(1,r)$ is a complete graph, thus it is clear to show the two-disjoint-cycle-cover pancyclicity in $D(1,r)$.

If $n=2$, there is no cycle of length 3 in $D(2,r)$.

If $n=3$ and $r=1$, there are no cycles of length ℓ with $𝓁=4,5$.

Other Combined Cases:

If $n\ge 3$ and $r\ge 2$, we obtain Theorem 4, which is the main result of this paper.

Theorem 4.

$D(n,r)$ is two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋]$-pancyclic, where $n\ge 3$ and $r\ge 2$.

The proof of Theorem 4 is primarily structured into two parts: Section 3 addresses the case $n=4$, while Section 4 generalizes the result to $n\ge 5$. For $n=3$, the proof approach is analogous to that for $n=4$, but to avoid redundancy, we computationally verify the two-disjoint-cycle-cover pancyclicity of $D(3,r)$ (with $r\ge 2$) using a Python implementation (with Python 3.12.7). This implementation (code available at https://github.com/guanlin-he/disjoint-cycle-pancyclicity-dragonfly (accessed on 14 November 2025)) leverages depth-first search and backtracking algorithms to exhaustively validate the property.

Our finding generalizes prior results in [2] (Theorems 1 and 2) about the Hamiltonicity and pancyclicity of $D(n,r)$. And we have the following corollary.

Corollary 1.

$D(n,r)$ is vertex-disjoint-cycle-coverable, where $n\ge 3$ and $r\ge 2$.

Table 1. Key notations used in this paper.

Notation	Meaning
$D(n,r$)	Dragonfly network with parameters n and r
n	Number of vertices in each group of $D(n,r)$
r	Number of external edges that each vertex has
g	Number of groups in $D(n,r)$ ($g=nr+1$)
$G_i$	A group of $D(n,r)$ with $i=0,1,\cdots ,g-1$
$K_n$	A complete graph consisting of n vertices

summarizes some key notations used throughout this paper.

3. Two-Disjoint-Cycle-Cover Pancyclicity in $\mathbf{D}(\mathbf{4},\mathbf{r})$ with $\mathbf{r}\ge \mathbf{2}$

This section focuses on proving the two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(4,r))|}{2}}⌋]$-pancy-clicity of $D(4,r)$ with $r\ge 2$ (formally presented by Theorem 5), which serves as a foundational case for the general result in Theorem 4. The proof proceeds by case analysis, covering different ranges of ℓ. Although the approach involves multiple cases, we strive to group similar scenarios and leverage the symmetric properties of $D(4,r)$ to reduce redundancy. Each case constructs cycles explicitly, using paths and edges derived from the graph’s topology. The detailed analysis here will provide a blueprint for extending the result to general $n\ge 5$ in Section 4.

Theorem 5.

For $r\ge 2$, $D(4,r)$ is two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(4,r))|}{2}}⌋]$-pancyclic.

Proof of Theorem 5.

Note that, by Definition 1, $|V(D(4,r))|=4g$ with $g=(4r+1)$. Thus, $⌊{\displaystyle \frac{|V(D(4,r))|}{2}}⌋=2g$. We only need to prove that $D(4,r)$ has two cycles $C_1$ and $C_2$ where $|V(C_1)|=𝓁$ and $|V(C_2)|=ng-𝓁$ with $3\le 𝓁\le 2g$.

For each ℓ within $3\le 𝓁\le 8$, we find the two cycles $C_1$ and $C_2$ presented in Table 2 and the corresponding figures, where $P_i$ denoted a Hamiltonian path between any two vertices in $G_i$ for $0\le i\le g-1$. For a cycle $C=c_1{c}_2\cdots c_p{c}_1$, we denote by $C[c_i,c_j]$ be a path between $c_i$ and $c_j$ in C. Let $\overline{C}=c_p{c}_{p-1}\dots c_1{c}_p$. Similarly, we denote by $P[p_1,p_q]=p_1{p}_2\cdots p_q$ and $\overline{P}[p_q,p_1]=p_q{p}_{q-1}\cdots p_1$.

For $9\le 𝓁\le 2g$, we present Claims 1–4 that declare the existence of two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ and prove their validity in turn. For clarity, we assign explicit symbols (see Table 3) to several special edges and sets (which certainly exist according to Definition 1) involved in the proofs of Claims 1–4. The proofs are constructive and share a common underlying idea. For each of Claims 1–4, we divide the proof into several cases based on the range of i. In each case, we first identify a Hamiltonian path $P_j$ in $G_j$, exploiting the isomorphism $G_j\cong K_n$ for each $j\in B_1$ or $j\in \{r-i,r-i+1,\cdots ,r\}$ (where $i\le r-1$). Subsequently, by removing specific vertices from groups $G_0$ and $G_{2r}$, we derive specific edges via Definition 1 and Theorem 3. These edges, combined with the paths $P_j$, form a cycle $C_1$. To obtain the other cycle $C_2$, we modify the Hamiltonian cycle $C_0$ by removing the vertices of $C_1$ and incorporate specific paths and edges, yielding a new cycle that satisfies the required length condition. The proof concludes by verifying the vertex counts and the disjointness of $C_1$ and $C_2$. Overall, the proof hinges on the combinatorial properties of $D(n,r)$ and a thorough case analysis encompassing all admissible values of ℓ.

Table 2. Examples of two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ for $3\le 𝓁\le 8$, where $𝓁=|V(C_1)|$.

ℓ	${\mathit{C}}_1$	${\mathit{C}}_2$	Figure
3	$((0,0)(0,1)(0,2)(0,0))$	$(P_1[(1,2),(1,0)]C_0\left[(2,3)(g-3,0)\right])$ $P_{g-1}[(g-1,3),(g-1,0)](0,3)$ $P_{g-2}[(g-2,0),(g-2,1)](1,2))$	Figure 3
4	$((0,0)(0,1)(0,2)(0,3)$ $(0,0))$	$(C_0[(1,3),(g-1,0)](1,3))$
5	$(((0,0)(1,3)(1,2)(1,0)$ $(2,3)(0,0))$	$(P_2[(2,2),(2,0)]C_0[(2,0),(r+1,3)]$ $P_{r+1}\left[(r+1,3)(r+1,2)\right](0,1)(0,2)(0,3)$ $\overline{C_0}[(0,3),(3r+2,0)]P_{3r+2}[(3r+2,0),(3r+2,2)]$ $P_{r+2,1}[(r+2,1),(r+2,2)](1,1)$ $P_{r+3}[(r+3,2),(r+3,0)]$ $C_0[(R+3,0),(3R+1,1)](2,2))$)	Figure 4
6	$((0,0)(r,3)(r,0)(2r,3)$ $(2r,2)(0,1)(0,0))$	$(C_0[(1,3),(r-1,0)](r+1,3)C_0[(r+1,3),(2r-1,0)]$ $(2r+1,3)C_0[(2r+1,3),(3r-1,3)]$ $P_{3r-1}[(3r-1,3),(3r-1,1)]$ $(0,2)(0,3)(4r-1,0)\overline{C_0}[(4r-1,0),(3r+1,0)]$ $P_{3r+1}[(3r+1,0),(3r+1,1)](r,2)(r,1)(3r,2)$ $P_{3r}[(3r,2),(3r,3)](2r,0)(2r,1)$ $(g-1,2)P_{g-1}[(g-1,2),(g-1,0)](1,3))$	Figure 5
7	$(0,0)(r,3)(r,0)(2r,3)$ $(2r,2)(0,1)(0,2)(0,0)$	$(C_0[(1,3),(r-1,0)](r+1,3)C_0[(r+1,3),(2r-1,0)]$ $(2r+1,3)P_{2r+1}[(2r+1,3),(2r+1,2)](r,1)$ $(r,2)(g-1,1)(g-1,2)(g-1,3)(4r-2,0)$ $\overline{C_0}[(4r-2,0),(2r+2,3)](2r,0)(2r,1)(4r-1,2)$ $P_{4r-1}[(4r-1,2),(4r-1,0)](0,3)(g-1,0)(1,3))$	Figure 6
8	$((0,0)(r,3)(r,2)(r,1)$ $(r,0)(2r,3)(2r,2)(0,1)$ $(0,0))$	$(C_0[(1,3),(r-1,0)](r+1,3)C_0[(r+1,3),(2r-1,0)]$ $(2r+1,3)P_{2r+1}[(2r+1,3),(2r+1,1)](0,2)(0,3)$ $(4r-1,0)\overline{C_0}[(4r-1,0),(2r+2,3)](2r,0)(2r,1)$ $(g-1,2)P_{g-1}[(g-1,2),(g-1,0)](1,3))$	Figure 7

Table 2. Examples of two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ for $3\le 𝓁\le 8$, where $𝓁=|V(C_1)|$.

ℓ	${\mathit{C}}_1$	${\mathit{C}}_2$	Figure
3	$((0,0)(0,1)(0,2)(0,0))$	$(P_1[(1,2),(1,0)]C_0\left[(2,3)(g-3,0)\right])$ $P_{g-1}[(g-1,3),(g-1,0)](0,3)$ $P_{g-2}[(g-2,0),(g-2,1)](1,2))$	Figure 3
4	$((0,0)(0,1)(0,2)(0,3)$ $(0,0))$	$(C_0[(1,3),(g-1,0)](1,3))$
5	$(((0,0)(1,3)(1,2)(1,0)$ $(2,3)(0,0))$	$(P_2[(2,2),(2,0)]C_0[(2,0),(r+1,3)]$ $P_{r+1}\left[(r+1,3)(r+1,2)\right](0,1)(0,2)(0,3)$ $\overline{C_0}[(0,3),(3r+2,0)]P_{3r+2}[(3r+2,0),(3r+2,2)]$ $P_{r+2,1}[(r+2,1),(r+2,2)](1,1)$ $P_{r+3}[(r+3,2),(r+3,0)]$ $C_0[(R+3,0),(3R+1,1)](2,2))$)	Figure 4
6	$((0,0)(r,3)(r,0)(2r,3)$ $(2r,2)(0,1)(0,0))$	$(C_0[(1,3),(r-1,0)](r+1,3)C_0[(r+1,3),(2r-1,0)]$ $(2r+1,3)C_0[(2r+1,3),(3r-1,3)]$ $P_{3r-1}[(3r-1,3),(3r-1,1)]$ $(0,2)(0,3)(4r-1,0)\overline{C_0}[(4r-1,0),(3r+1,0)]$ $P_{3r+1}[(3r+1,0),(3r+1,1)](r,2)(r,1)(3r,2)$ $P_{3r}[(3r,2),(3r,3)](2r,0)(2r,1)$ $(g-1,2)P_{g-1}[(g-1,2),(g-1,0)](1,3))$	Figure 5
7	$(0,0)(r,3)(r,0)(2r,3)$ $(2r,2)(0,1)(0,2)(0,0)$	$(C_0[(1,3),(r-1,0)](r+1,3)C_0[(r+1,3),(2r-1,0)]$ $(2r+1,3)P_{2r+1}[(2r+1,3),(2r+1,2)](r,1)$ $(r,2)(g-1,1)(g-1,2)(g-1,3)(4r-2,0)$ $\overline{C_0}[(4r-2,0),(2r+2,3)](2r,0)(2r,1)(4r-1,2)$ $P_{4r-1}[(4r-1,2),(4r-1,0)](0,3)(g-1,0)(1,3))$	Figure 6
8	$((0,0)(r,3)(r,2)(r,1)$ $(r,0)(2r,3)(2r,2)(0,1)$ $(0,0))$	$(C_0[(1,3),(r-1,0)](r+1,3)C_0[(r+1,3),(2r-1,0)]$ $(2r+1,3)P_{2r+1}[(2r+1,3),(2r+1,1)](0,2)(0,3)$ $(4r-1,0)\overline{C_0}[(4r-1,0),(2r+2,3)](2r,0)(2r,1)$ $(g-1,2)P_{g-1}[(g-1,2),(g-1,0)](1,3))$	Figure 7

Figure 3. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=3$.

Figure 3. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=3$.

Figure 4. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=5$.

Figure 4. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=5$.

Figure 5. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=6$.

Figure 5. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=6$.

Figure 6. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=7$.

Figure 6. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=7$.

Figure 7. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=8$.

Figure 7. Two vertex-disjoint cycles $C_1$ (in red) and $C_2$ (in black) in $D(4,r)$ for $|V(C_1)|=𝓁=8$.

Table 3. Key symbols used in the proofs of Claims 1–4.

Symbols and Definitions
$e^1=((r,0),(2r-[i-(r-1)],3))$ with $r\le i\le 2r-3$
$e^2=((r-i-1,0),(r+1,3))$ with $0\le i\le r-2$
$e^3=((2r,0),(2r+2,3))$
$e^4=((1,3),(g-1,0))$
$e^5=((2r-1,0),(2r+1,3))$
$e^6=((2r-[i-(r-1)]-1,0),(2r+1,3))$ with $r-1\le i\le 2r-3$
$e^7=((g-1,1),(r+1,2))$
$B_1=\{1,2,\cdots ,2r-1\}/\{r+1,r+2,\cdots ,2r-[i-(r-1)]-1\}$ with $r\le i\le 2r-3$ and $|B_1|=i+1$
$B_2=\{0,4r,2r+1,2r,4r-1\}$
$B_3=\{0,2r+1,2r\}$

Table 3. Key symbols used in the proofs of Claims 1–4.

Symbols and Definitions
$e^1=((r,0),(2r-[i-(r-1)],3))$ with $r\le i\le 2r-3$
$e^2=((r-i-1,0),(r+1,3))$ with $0\le i\le r-2$
$e^3=((2r,0),(2r+2,3))$
$e^4=((1,3),(g-1,0))$
$e^5=((2r-1,0),(2r+1,3))$
$e^6=((2r-[i-(r-1)]-1,0),(2r+1,3))$ with $r-1\le i\le 2r-3$
$e^7=((g-1,1),(r+1,2))$
$B_1=\{1,2,\cdots ,2r-1\}/\{r+1,r+2,\cdots ,2r-[i-(r-1)]-1\}$ with $r\le i\le 2r-3$ and $|B_1|=i+1$
$B_2=\{0,4r,2r+1,2r,4r-1\}$
$B_3=\{0,2r+1,2r\}$

Claim 1.

For $𝓁=9+4i, i=0,1,\cdots ,\lfloor {\displaystyle \frac{(2g-8)}{4}}\rfloor$, two vertex-disjoint cycles $C_1$, $C_2$ can be found in $D(4,r)$, where $|V(C_1)|=𝓁$, $|V(C_2)|=4g-𝓁$.

Proof of Claim 1.

It follows from the condition $𝓁\le 2g$ in $D(4,r)$ that $i\le 2r-2$. For $j=0,1,\cdots ,g-1$, since $G_j$ is a complete graph, there is a Hamiltonian path $P_j$ between $(j,0)$ and $(j,n-1)$ in $G_j$.

There are two paths: $P_1^{\prime }=(r-i,3)(0,0)(0,2)(0,1)(2r,2)(2r,3)(r,0)$ with $|V(P_1^{\prime })|=7$ and $P_1^{″}=(1,3)(0,0)(0,2)(0,1)(2r,2)(2r,3)(2r-1,0)$ with $|V(P_1^{″})|=7$ such that $|V(P_1^{\prime }\cap ({\bigcup }_{m=r-i}^r{P}_m))|=2$ and $|V(P_1^{″}\cap ({\bigcup }_{m\in B_1}{P}_m))|=2$. There exists a cycle, $C_1$, of length ℓ:

$$C_1=\left\{\begin{array}{cc}(\bigcup _{m=r-i}^r{P}_m)\cup P_1^{\prime }\cup (\bigcup _{a=r-i}^r\left\{((a,0),(a+1,3))\right\}),\hfill & if i\le r-1;\hfill \\ \\ (\bigcup _{m\in B_1}{P}_m)\cup P_1^{\prime \prime }\cup (\bigcup _{a\in B_1}\left\{((a,0),(a+1,3))\right\})\cup \left\{e^1\right\},\hfill & if i\ge r.\hfill \end{array}\right.$$

where $|V(C_1)|=4i+9$ since $V(e^1)\subseteq {\bigcup }_{m\in B_1}{P}_m$ and

$$|V(C_1)|=\left\{\begin{array}{cc}\sum _{m=r-i}^r|V(P_m)|+|V(P_1^{\prime })|-|V(P_1^{\prime }\cap (\bigcup _{m=r-i}^r{P}_m))|,\hfill & if i\le r-1;\hfill \\ \\ |B_1||V(P_m)|+|V(P_1^{\prime \prime })|-|V(P_1^{\prime \prime }\cap (\bigcup _{m\in B_1}{P}_m))|,\hfill & if i\ge r.\hfill \end{array}\right.$$

According to the range of i, we can obtain a cycle $C_2$ of length $4g-𝓁$:

When $i\le r-2$, there are two paths $P_1^1=(1,3)(g-1,0)(0,3)(4r-1,0)(4r-1,3)$$(4r-1,1)(4r-1,2)(2r,1)(2r,0)(2r+2,3)$ and $P_1^2=(2r-1,0)(2r+1,3)(2r+1,2)$$(2r+1,0)(2r+1,1)(g-1,2)(g-1,1)(g-1,3)(4r-2,0)$ in $D(4,r)$ such that $|V(P_1^1)|=10$ and $|V(P_1^2)|=9$. Then

$$C_2=(C_0-(\bigcup _{a\in B_2}V(G_a)\cup \bigcup _{m=r-i}^{r}V(G_m)))\cup P_1^1\cup P_1^2\cup \left\{e^2\right\}.$$

In order to facilitate the calculation of the length of the constructed cycle $C_2$, we introduce the following notation. Let $F_1=C_0-({\bigcup }_{a\in B_2}V(G_a)\cup {\bigcup }_{m=r-i}^{r}V(G_m))$. We have $V(e^2)\subseteq V(F_1)$ and $|V(P_1^{\alpha })\cap V(F_1)|=2$ with $\alpha =1,2$. So,

$$\begin{array}{cc}\hfill |V(C_2)|& =|V(F_1)|+|V(P_1^1)|+|V(P_1^2)|-\sum _{\alpha =1}^2|V(P_1^{\alpha }\cap F_1)|\hfill \\ & =4g-|B_2||V(G_j)|-\sum _{m=r-i}^r|V(G_m)|+10+9-4\hfill \\ & =4g-5\times 4-4(i+1)+15=4g-4i-9.\hfill \end{array}$$

When $r-1\le i\le 2r-3$, there are paths $P_1^3=(2r,1)(4r-1,2)(4r-1,1)(4r-1,3)(4r-1,0)(0,3)(g-1,0)(g-1,1)(r+1,2)(r+1,1)(r+1,0)(r+1,3)(r+2,0)$ with $|V(P_1^3)|=13$ and $P_1^4=(2r-[i-(r-1)]-1,0)(2r+1,3)(2r+1,2)(2r+1,0)(2r+1,1)(g-1,2)(g-1,3)(4r-2,0)$ with $|V(P_1^4)|=8$ in $D(4,r)$. By Theorem 3, let $e^3=((2r,0),(2r+2,3))$. Thus,

$$\begin{array}{c}\hfill C_2=(C_0-(\bigcup _{a=2r-[i-(r-1)]}^{2r+1}V(G_a)\cup \bigcup _{m=0}^{r+1}V(G_m))\cup V(G_{g-1})\cup V(G_{g-2}))\cup P_1^3\cup P_1^4\cup \left\{e^3\right\}\end{array}$$

Similarly, let $F_2=C_0-({\bigcup }_{a=2r-[i-(r-1)]}^{2r+1}V(G_a)\cup {\bigcup }_{m=0}^{r+1}V(G_m))\cup V(G_{g-1})\cup V(G_{g-2})$. We have $|V(P_1^3\cap F_2)|=1$, $|V(P_1^4\cap F_2)|=2$ and $|V(e^3\cap F_2)|=1$. Then,

$$\begin{array}{cc}\hfill |V(C_2)|& =|V(F_2)|+|V(P_1^3)|+|V(P_1^4)|-\sum _{\alpha =3}^4|V(P^{\alpha }\cap F_2)|\hfill \\ & =4g-|B_2||V(G_j)|-\sum _{m=r-i}^r|V(G_m)|+10+9-4\hfill \\ & =4g-5\times 4-4(i+1)+15=4g-4i-9.\hfill \end{array}$$

When $i=2r-2$, there is a path $P_1^5=(2r,0)(2r,1)(g-1,2)(g-1,3)(g-1,1)(g-1,0)(0,3)(4r-1,0)$ with $|V(P_1^5)|=8$ and $|V(P_1^5\cap (C_0-({\bigcup }_{m=0}^{2r}V(G_m))\cup V(G_{g-1}))))|=1$ in $D(4,r)$. Thus,

$$C_2=(C_0-(\bigcup _{m=0}^{2r}V(G_m))\cup V(G_{g-1})))\cup P_1^5,$$

where

$$\begin{array}{ccc}\hfill |V(C_2)|& =\hfill & |V((C_0-(\bigcup _{m=0}^{2r}V(G_m))\cup V(G_{g-1}))))|+|V(P_1^5)|\hfill \\ & & -|V(P_1^5\cap (C_0-(\bigcup _{m=0}^{2r}V(G_m))\cup V(G_{g-1}))))|\hfill \\ & =\hfill & 4g-(2r+2))|V(G_j)|+8-1\hfill \\ & =\hfill & 4g-8r-1=4g-4i-9.\hfill \end{array}$$

So, $C_1$, $C_2$ are two disjoint cycles of $D(4,r)$, where $|C_1|=𝓁$, $|C_2|=4g-𝓁$ (see Figure 8, Figure 9 and Figure 10).

Figure 8. Two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ for $i\le r-2$ in Claim 1. The red edges form a cycle $C_1$, while the combination of green, blue, and black edges forms a cycle $C_2$.

Figure 8. Two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ for $i\le r-2$ in Claim 1. The red edges form a cycle $C_1$, while the combination of green, blue, and black edges forms a cycle $C_2$.

Figure 9. Two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ for $r-1\le i\le 2r-3$ in Claim 1. The red edges form a cycle $C_1$, while the combination of green, blue, and black edges forms a cycle $C_2$.

Figure 9. Two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ for $r-1\le i\le 2r-3$ in Claim 1. The red edges form a cycle $C_1$, while the combination of green, blue, and black edges forms a cycle $C_2$.

Figure 10. Two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ for $i=2r-2$ in Claim 1. The red edges form a cycle $C_1$, while the combination of green and black edges forms a cycle $C_2$.

Figure 10. Two vertex-disjoint cycles $C_1$ and $C_2$ in $D(4,r)$ for $i=2r-2$ in Claim 1. The red edges form a cycle $C_1$, while the combination of green and black edges forms a cycle $C_2$.

□

Using the same argument as for Claim 1, we obtain the following claims.

Claim 2.

For $𝓁=10+4i, i=0,1,\cdots ,\lfloor {\displaystyle \frac{2g-9}{4}}\rfloor$, two vertex-disjoint cycles $C_1$, $C_2$ can be found in $D(4,r)$, where $|V(C_1)|=𝓁$, $|V(C_2)|=4g-𝓁$.

Proof of Claim 2.

Since $G_j$ is a complete graph, for $j=0,1,\cdots ,g-1$, there is a Hamiltonian path $P_j$ between $(j,0)$ and $(j,n-1)$ in $G_j$.

We have two paths: $P_2^{\prime }=(r-i,3)(0,0)(0,2)(0,3)(0,1)(2r,2)(2r,3)(r,0)$ with $|V(P_2^{\prime })|=8$, and $P_2^{″}=(1,3)(0,0)(0,2)(0,3)(0,1)(2r,2)(2r,3)(2r-1,0)$ with $|V(P_2^{″})|=8$. We have $|V(P_2^{\prime }\cap {\bigcup }_{m=r-i}^r{P}_m)|=2$ and $|V(P_2^{″}\cap {\bigcup }_{m\in B_1}{P}_m)|=2$.

There is a cycle $C_1$ of length ℓ:

$$C_1=\left\{\begin{array}{cc}(\bigcup _{m=r-i}^r{P}_m)\cup P_2^{\prime }\cup (\bigcup _{a=r-i}^r\left\{((a,0),(a+1,3))\right\}),\hfill & if i\le r-1;\hfill \\ \\ (\bigcup _{m\in B_1}{P}_m\cup P_2^{\prime \prime }\cup (\bigcup _{a\in B_1}\left\{((a,0),(a+1,3))\right\})\cup \left\{e^1\right\},\hfill & if i\ge r.\hfill \end{array}\right.$$

where $|V(C_1)|=4i+10$ since $V(e^1)\subseteq {\bigcup }_{m\in B_1}{P}_m$ and

$$|V(C_1)|=\left\{\begin{array}{cc}\sum _{m=r-i}^r|V(P_m)|+|V(P_2^{\prime })|-|V(P_2^{\prime }\cap (\bigcup _{m=r-i}^r{P}_m))|,\hfill & if i\le r-1;\hfill \\ \\ |B_1||V(P_m)|+|V(P_2^{\prime \prime })|-|V{(P)}_2^{\prime \prime }\cap (\bigcup _{m\in B_1}{P}_m))|,\hfill & if i\ge r.\hfill \end{array}\right.$$

According to the range of i, we can obtain a cycle $C_2$ of length $4g-𝓁$:

If $i\le r-2$, by Theorem 3, there is a path $P_2^1=(2r-1,0)(2r+1,3)(2r+1,2)(2r+1,0)(2r+1,1)(4r-1,2)(2r,1)(2r,0)(2r+2,3)$ with $|V(P_2^1)|=9$ in $D(4,r)$. Since $G_i\cong K_n$, we have $(4r-1,1),(4r-1,3))\in E(D(n,r))$. Let $(4r-1,1),(4r-1,3))=e_2^1$. Thus,

$$C_2=(C_0-(\bigcup _{a\in B_3}V(G_a)\cup \bigcup _{m=r-i}^{r}V(G_m)\cup \left\{(4r-1,2)\right\}))\cup P_2^1\cup \{e^2,e^4,e_2^1\}.$$

In order to facilitate the calculation of the length of the constructed cycle $C_2$, we introduce the following notation. Let $F_3=C_0-({\bigcup }_{a\in B_3}V(G_a)\cup {\bigcup }_{m=r-i}^{r}V(G_m)\cup \left\{(4r-1,2)\right\})$. We have $|V(P_1^1\cap F_3)|=2$ and $V(\{e^2,e^4,e_2^1\})\subseteq V(F_3)$. So,

$$\begin{array}{cc}\hfill |V(C_2)|& =|V(F_3)|+|V(P_2^1)|-|V(P_2^1\cap F_3)|\hfill \\ & =4g-|B_3||V(G_j)|-\sum _{m=r-i}^r|V(G_m)|-1+9-2\hfill \\ & =4g-3\times 4-4(i+1)+6=4g-4i-10.\hfill \end{array}$$

If $r-1\le i\le 2r-3$, there are paths $P_2^2=(2r-[i-(r-1)]-1,0)(2r+1,3)(2r+1,2)(2r+1,0)(2r+1,1)(4r-1,2)(2r,1)(2r,0)(2r+2,3)$ and $P_2^3=(g-1,1)(r+1,2)(r+1,1)(r+1,0)(r+1,3)$ in $D(4,r)$ with $|V(P_2^2)|=9$ and $|V(P_2^3)|=5$. Then

$$\begin{array}{cc}\hfill C_2=& (C_0-(\bigcup _{a=2r-[i-(r-1)]}^{2r+1}V(G_a)\cup \bigcup _{m=0}^{r}V(G_m))\cup V(G_{g-2}))\cup P_2^2\cup P_2^3\cup \left\{e_2^1\right\}\hfill \end{array}$$

Similarly, let $F_4=(C_0-({\bigcup }_{a=2r-[i-(r-1)]}^{2r+1}V(G_a)\cup {\bigcup }_{m=0}^{r}V(G_m))\cup V(G_{g-2})$. We have $|V(P_2^2\cap F_4)|=3$ and $|V(P_2^3\cap F_4)|=1$, $V(e_2^1)\subseteq V(F_4)$. So,

$$\begin{array}{cc}\hfill |V(C_2)|& =|V(F_4)|+|V(P_2^2)|+|V(P_2^3)|-\sum _{\alpha =2}^3|V(P_2^{\alpha }\cap F_4)|\hfill \\ & =4g-(i-r+3)|V(G_a)|-\sum _{m=0}^r|V(G_m)|\hfill \\ & -|V(G_{g-2})|+9+5-4\hfill \\ & =4g-4(i+5)+10=4g-4i-10.\hfill \end{array}$$

If $i=2r-2$, there is a path $P_2^4=(2r,0)(2r,1)(g-1,2)(g-1,1)(g-1,3)(g-1,0)$ with $|V(P_2^4)|=6$ in $D(4,r)$ by Theorem 3. Thus,

$$C_2=(C_0-(\bigcup _{m=0}^{2r-1}V(G_m))\cup V(G_{g-1}))\cup P_2^4.$$

Similarly, let $F_5=(C_0-({\bigcup }_{m=0}^{2r-1}V(G_m))\cup V(G_{g-1}))$. We have $|V(P_2^4\cap F_5)|=4$. Thus,

$$\begin{array}{cc}\hfill |V(C_2)|& =|V(F_5)|+|V(P_2^4)|-|V(P_2^4\cap F_5)|\hfill \\ & =4g-(2r+2)|V(G_a)|+6-4\hfill \\ & =4g-8r-2=4g-4i-10.\hfill \end{array}$$

Then $C_1$ and $C_2$ are two disjoint cycles of $D(4,r)$, where $|C_1|=𝓁$, $|C_2|=4g-𝓁$. □

Claim 3.

For $𝓁=11+4i, i=0,1,\cdots ,\lfloor {\displaystyle \frac{2g-10}{4}}\rfloor -1$, two vertex-disjoint cycles $C_1$, $C_2$ can be found in $D(4,r)$, where $|V(C_1)|=𝓁$, $|V(C_2)|=4g-𝓁$.

Proof of Claim 3.

It follows from $𝓁\le 2g$ in $D(4,r)$ that $i\le 2r-3$. Since $G_j$ is a complete graph, for $j=0,1,\cdots ,g-1$, there is a Hamiltonian path $P_j$ between $(j,0)$ and $(j,n-1)$ in $G_j$. There are two paths: $P_3^{\prime }=(r-i,3)(0,0)(0,2)(0,1)(2r,2)(2r,1)(2r,0)(2r,3)(r,0)$, and $P_3^{″}=(1,3)(0,0)(0,2)(0,1)(2r,2)(2r,1)(2r,0)(2r,3)(2r-1,0)$. We have $|V(P_3^{\prime })|=9$, $|V(P_3^{″})|=9$, $|V(P_3^{\prime }\cap {\bigcup }_{m=r-i}^r{P}_m)|=2$ and $|V(P_3^{″}\cap {\bigcup }_{m\in B_1}{P}_m)|=2$. There is a cycle of length ℓ:

$$C_1=\left\{\begin{array}{cc}(\bigcup _{a=r-i}^r\left\{((a,0),(a+1,3))\right\})\cup (\bigcup _{m=r-i}^r{P}_m)\cup P_3^{\prime },\hfill & if i\le r-1;\hfill \\ \\ (\bigcup _{m\in B_1}{P}_m\cup P_3^{\prime \prime }\cup \left\{e^1\right\}\cup (\bigcup _{a\in B_1}\left\{((a,0),(a+1,3))\right\}),\hfill & if i\ge r.\hfill \end{array}\right.$$

where $|V(C_1)|=4i+11$ since

$$|V(C_1)|=\left\{\begin{array}{cc}\sum _{m=r-i}^r|V(P_m)|+|V(P_3^{\prime })|-|V(P_3^{\prime }\cap (\bigcup _{m=r-i}^r{P}_m))|,\hfill & if i\le r-1;\hfill \\ \\ |B_1||V(P_m)|+|V(P_3^{\prime \prime })|-|V(P_3^{\prime \prime }\cap (\bigcup _{m\in B_1}{P}_m))|,\hfill & if i\ge r.\hfill \end{array}\right.$$

According to the range of i, we can obtain a cycle $C_2$ of length $4g-𝓁$:

When $i\le r-2$, there are paths $P_3^1=(1,3)(g-1,0)(0,3)(4r-1,0)(4r-1,1)(4r-1,2)(4r-1,3)(3r-1,0)(3r,3)$ with $|V(P_3^1)|=9$ and $P_3^2=(3r-2,0)(3r-1,3)(3r-1,2)(3r-1,1)(g-1,2)(g-1,1)(g-1,3)(4r-2,0)$ with $|V(P_3^2)|=8$ in $D(4,r)$. Thus,

$$C_2=(C_0-(\bigcup _{a\in \{0,4r,3r-1,2r,4r-1\}}V(G_a)\cup \bigcup _{m=r-i}^{r}V(G_m)))\cup P_3^1\cup P_3^2\cup \{e^2,e^5\}.$$

In order to facilitate the calculation of the length of the constructed cycle $C_2$, we introduce the following notation. Let $F_6=C_0-({\bigcup }_{a\in \{0,4r,3r-1,2r,4r-1\}}V(G_a)\cup {\bigcup }_{m=r-i}^{r}V(G_m))$. Then $|V(P_3^{\beta }\cap F_6)|=2$ with $\beta =1,2$ and $V(e^2,e^5)\subseteq V(F_6)$. We have

$$\begin{array}{cc}\hfill |V(C_2)|& =|V(F_6)|+|V(P_3^1)+|V(P_3^2)|-\sum _{\beta =1}^2|V(P_3^{\beta }\cap F_6)|\hfill \\ & =4g-5|V(G_j)|-\sum _{m=r-i}^r|V(G_m)|+9+8-2-2\hfill \\ & =4g-4(i+6)+17-4=4g-4i-11.\hfill \end{array}$$

When $r-1\le i\le 2r-3$, there is a path $P_3^3=(4r-1,0)(0,3)(g-1,0)(g-1,3)$$(g-1,2)(g-1,1)(r+1,2)(r+1,0)(r+1,1)(r+1,3)(r+2,0)$ with $|V(P_3^3)|=11$ in $D(4,r)$. Thus,

$$C_2=(C_0-(\bigcup _{a=2r-[i-(r-1)]}^{2r}V(G_a)\cup \bigcup _{m=0}^{r}V(G_m)\cup V(G_{g-1}))))\cup P_3^3\cup \left\{e^6\right\}.$$

Similarly, let $F_7=C_0-({\bigcup }_{a=2r-[i-(r-1)]}^{2r}V(G_a)\cup {\bigcup }_{m=0}^{r}V(G_m)\cup V(G_{g-1})))$. We have $|V(P_3^3\cap F_7)|=2$ and $V(e^6)\subseteq V(F_7)$. So,

$$\begin{array}{cc}\hfill |V(C_2)|& =|V(F_7)|+|V(P_3^3)|+|V(P_3^3\cap F_7)|\hfill \\ & =4g-(i+5)|V(G_j)|+11-2\hfill \\ & =4g-4i-11.\hfill \end{array}$$

So, $C_1$ and $C_2$ are two disjoint cycles of $D(4,r)$, where $|V(C_1)|=𝓁$, $|V(C_2)|=4g-𝓁$. □

Claim 4.

For $𝓁=12+4i, i=0,1,\cdots ,\lfloor {\displaystyle \frac{2g-11}{4}}\rfloor -1$, two vertex-disjoint cycles $C_1$, $C_2$ can be found in $D(4,r)$, where $|V(C_1)|=𝓁$, $|V(C_2)|=4g-𝓁$.

Proof of Claim 4.

It follows from $𝓁\le 2g$ in $D(4,r)$ that $i\le 2r-3$. Since $G_j$ is a complete graph, for $j=0,1,\cdots ,g-1$, there is a Hamiltonian path $P_j$ between $(j,0)$ and $(j,n-1)$ in $G_j$.

There are two paths: $P_4^{\prime }=(r-i,3)(0,0)(0,2)(0,3)(0,1)(2r,2)(2r,1)(2r,2)(2r,3)(r,0)$, and $P_4^{″}=(1,3)(0,0)(0,2)(0,3)(0,1)(2r,2)(2r,1)(2r,0)(2r,3)(2r-1,0)$. We have $|V(P_4^{\prime })|=|V(P_4^{\prime \prime })|=10$ and $|V(P_4^{\prime }\cap ({\bigcup }_{m=r-i}^r{P}_m))|=|P_4^{″}\cap ({\bigcup }_{m\in B_1}{P}_m)|=2$. So, there is a cycle $C_1$ of length ℓ:

$$C_1=\left\{\begin{array}{cc}(\bigcup _{a=r-i}^r\left\{((a,0),(a+1,3))\right\})\cup (\bigcup _{m=r-i}^r{P}_m)\cup P_4^{\prime },\hfill & if i\le r-1\hfill \\ \\ (\bigcup _{a\in B_1}\left\{((a,0),(a+1,3))\right\})\cup \left\{e^1\right\}\cup (\bigcup _{m\in B_1}{P}_m)\cup P_4^{\prime \prime }\hfill & if i\ge r.\hfill \end{array}\right.$$

where $|V(C_1)|=4i+12$ since

$$|V(C_1)|=\left\{\begin{array}{cc}\sum _{m=r-i}^r|V(P_m)|+|V(P_4^{\prime })|-|V(P_4^{\prime }\cap (\bigcup _{m=r-i}^r{P}_m))|,\hfill & if i\le r-1\hfill \\ \\ |B_1||V(P_m)|+|V(P_4^{\prime \prime })|-|V(P_4^{\prime \prime }\cap (\bigcup _{m\in B_1}{P}_m))|,\hfill & if i\ge r.\hfill \end{array}\right.$$

According to the range of i, we can obtain a cycle $C_2$ of length $4g-𝓁$:

When $i\le r-2$, we have the following cycle $C_2$

$$C_2=(C_0-(\bigcup _{a\in \{0,2r\}}V(G_a)\cup \bigcup _{m=r-i}^{r}V(G_m)))\cup \{e^2,e^4,e^5\}.$$

In order to facilitate the calculation of the length of the constructed cycle $C_2$, we introduce the following notation. Let $F_8=C_0-({\bigcup }_{a\in \{0,2r\}}V(G_a)\cup {\bigcup }_{m=r-i}^{r}V(G_m))$. Then $V(\{e^2,e^4,e^5\})\subseteq F_8$. We have

$$\begin{array}{c}\hfill |V(C_2)|=|V(F_8)|=4g-2|V(G_j)|-\sum _{m=r-i}^r|V(G_m)|=4g-4(i+6)=4g-4i-12.\end{array}$$

When $r-1\le i\le 2r-3$, $V(\{e^6,e^7\})\subseteq (C_0-({\bigcup }_{a=2r-[i-(r-1)]}^{2r}V(G_a))$. Thus,

$$C_2=(C_0-(\bigcup _{a=2r-[i-(r-1)]}^{2r}V(G_a))\cup \bigcup _{m=0}^{r}V(G_m)))\cup \{e^6,e^7\}.$$

where

$$\begin{array}{cc}\hfill |V(C_2)|& =4g-\sum _{a=2r-[i-(r-1)]}^{2r}|V(G_a)|-\sum _{m=r-i}^r|V(G_m)|\hfill \\ & =4g-4(i-r+2)-4(r+1)=4g-4i-12.\hfill \end{array}$$

So, $C_1$ and $C_2$ are two disjoint cycles of $D(4,r)$, where $|V(C_1)|=𝓁$, $|V(C_2)|=4g-𝓁$. □

By the above case analysis, two vertex-disjoint cycles $C_1$, $C_2$ can be found in $D(4,r)$ with $|V(C_1)|=𝓁$, $|V(C_2)|=ng-𝓁$, where $3\le 𝓁\le 2g$. □

4. Two-Disjoint-Cycle-Cover Pancyclicity in $\mathit{D}(\mathit{n},\mathit{r})$ with $\mathit{n}\ge \mathbf{5}$ and $\mathit{r}\ge \mathbf{2}$

In this section, we focus on proving the two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋]$-pancyclicity of $D(n,r)$ with $n\ge 5$ and $r\ge 2$, extending the result from Section 3 to prove Theorem 4. The proof builds on the techniques developed for $D(4,r)$ but adapts them to handle the cases of larger n. We partition the range of ℓ into three main cases (Case 1: $3\le 𝓁\le 5$; Case 2: $6\le 𝓁\le 3n-2$; Case 3: $3n-1\le 𝓁\le \lfloor {\displaystyle \frac{ng}{2}}\rfloor$) to systematically address all possible cycle lengths. Each case leverages the Hamiltonian cycle $C_0$ (defined in Section 2) and the complete graph structure of the groups $G_i$. By combining path constructions, edge manipulations, and inductive arguments, we demonstrate the existence of the required cycles $C_1$ and $C_2$ for every ℓ. While the proof involves case analysis, we emphasize the unifying principles, such as the use of symmetry and recursive group-based constructions, to provide insight into the general problem. This structured approach ensures rigor while minimizing ad hoc computations.

Case 1.

$3\le 𝓁\le 5$.

Since $G_i\cong K_n$ with $0\le i\le g-1$, so $G_i$ contains a cycle of length ℓ, where $n\ge 5$. As $D(n,r)$ possesses the property of rotational symmetry, we can pick group 0 as a representative case without loss of generality. Let $C_1=((0,1)(0,2)(0,3)(0,1))$ such that $|C_1|=3$ and $C_2=(C_0\backslash V(C_1))\cup \left\{((0,0),(0,4))\right\}$ such that $|C_2|=ng-3$. Now, we show $𝓁=4$. There is a cycle of length 4:

$$C_1^{\prime }=\left\{\begin{array}{cc}((0,0)(0,2)(0,3)(0,4)(0,0)),\hfill & if n=5\hfill \\ \\ ((0,1)(0,2)(0,3)(0,4)(0,1)),\hfill & if n\ge 6.\hfill \end{array}\right.$$

Since $G_i\cong K_n$ for every i in the range $0\le i\le g-1$, there exist Hamiltonian paths as follows: a path $P_1$ in $G_{r+1}$ connecting $(r+1,n-1)$ to $(r+1,n-2)$, and a path $P_2$ in $G_{r+2}$ connecting $(r+2,0)$ to $(r+2,n-2)$. When $n\ge 6$, there is a Hamiltonian path $P_3$ between $(0,0)$ and $(0,n-1)$ in $V(G_0)\backslash V(C_1^{\prime })$. By Lemma 1, we have $(0,1)\in N((r+1,n-2))\cap N((r+2,n-2))$. Let $P=(r+1,n-2)(0,1)(r+2,n-2))$. Then, we have $|V(P\cap P_1)|=|V(P\cap P_2)|=1$. By Definition 1 and $r\ge 2$, we have the following edges: $e_1=((r,0),(r+1,n-1))$, $e_2=((r+2,0),(r+3,n-1))$, $e_3=((1,n-1),(g-1,0))$.

Thus, there is a cycle $C^{\prime }$ of length $ng-4$:

$$C_2^{\prime }=\left\{\begin{array}{cc}(C_0-(\bigcup _{a\in \{0,r+1,r+2\}}V(G_a))\cup P\cup P_1\cup P_2\cup \{e_1,e_2,e_3\},\hfill & if n=5\hfill \\ \\ (C_0-V(G_0))\cup P_3\hfill & if n\ge 6.\hfill \end{array}\right.$$

where

$$|V(C_2^{\prime })|=\left\{\begin{array}{c}|V(C_0)|-3|V(G_a)|+|V(P)|+|V(P_1)|+|V(P_2)|\hfill \\ -|V(P\cap P_1)|-|V(P\cap P_2)|=5g-4,\hfill & if i\le r-1\hfill \\ \\ |V(C_0)|-|V(G_0)|+|V(P_3)=ng-n+(n-4)=ng-4,\hfill & if i\ge r.\hfill \end{array}\right.$$

So $C_1^{\prime }$, $C_2^{\prime }$ are proved to be two disjoint cycles of $D(n,r)$, where $|C_1^{\prime }|=4$, $|C_2^{\prime }|=ng-4$.

If $n=5$ and $𝓁=5$, by Lemma 2, then there are two disjoint cycles, one of which has a length of 5 and the other has a length of $ng-5$.

If $n\ge 6$ and $𝓁=5$, there is a cycle of length 5:

$$C_1^{\prime \prime }=\left\{\begin{array}{cc}((0,0)(0,2)(0,3)(0,4)(0,5)(0,0)),\hfill & if n=6.\hfill \\ \\ ((0,1)(0,2)(0,3)(0,4)(0,5)(0,1)),\hfill & if n\ge 7.\hfill \end{array}\right.$$

When $n\ge 7$, there is a Hamiltonian path $P_4$ between $(0,0)$ and $(0,n-1)$ in $V(G_0)-V(C_1^{\prime \prime })$.

There is a cycle of length $ng-𝓁$:

$$C_2^{\prime }=\left\{\begin{array}{cc}(C_0-(\bigcup _{a\in \{0,r+1,r+2\}}V(G_a))\cup P\cup P_1\cup P_2\cup \{e_1,e_2,e_3\},\hfill & if n=6\hfill \\ \\ (C_0-V(G_0))\cup P_4\hfill & if n\ge 7.\hfill \end{array}\right.$$

where

$$|V(C_2^{\prime })|=\left\{\begin{array}{c}|V(C_0)|-3|V(G_a)|+|V(P)|+|V(P_1)|+|V(P_2)|\hfill \\ -|V(P\cap P_1)|-|V(P\cap P_2)|=6g-5,\hfill & if i\le r-1\hfill \\ \\ |V(C_0)|-|V(G_0)|+|V(P_3)=ng-n+(n-5)=ng-5,\hfill & if i\ge r.\hfill \end{array}\right.$$

Case 2.

$6\le 𝓁\le 3n-2.$

By Definition 1 and $r\ge 2$, we have the following edges: $f_1=((0,n-3),((n-2)r-1,2))$, $f_2=(((n-1)r,n-2),((n-2)r-1,1))$ and $f_3=((0,n-2),((n-1)r,1))$. Then there is a cycle of length 6: $C=(0,n-3)((n-2)r-1,2)((n-2)r-1,1)((n-1)r,n-2)((n-1)r,1)(0,n-2)(0,n-3)$ in $G_0\cup G_{(n-1)r}\cup G_{(n-2)r-1}$ containing $f_1$,$f_2$ and $f_3$.

Since $G_i$ is a complete graph, there are some edges $e_1^{\prime }=((0,n-3),(0,n-2))$, $e_1^{\prime \prime }=((0,0),(0,n-1))$,$e_2^{\prime }=(((n-2)r-1,2),((n-2)r-1,1))$, $e_2^{\prime \prime }=(((n-2)r-1,0),((n-2)r-1,n-1))$, $e_3^{\prime }=(((n-1)r,n-2),((n-1)r,1))$ and $e_3^{\prime \prime }=(((n-1)r,n-1),((n-1)r,0))$. Let $V^{\prime }={\bigcup }_{i\in \{0,(n-1)r,((n-2)r-1)\}}V(G_i)\backslash \{(i,n-1),(i,0)\}$. For convenience, let $C^{\prime }=(C_0-V^{\prime })\cup \{e_1^{\prime \prime },e_2^{\prime \prime },e_3^{\prime \prime }\}$ such that $|C^{\prime }|=ng-3n+6$.

Since $G_i\cong K_n$ for every i in the range $0\le i\le g-1$, there exist paths as follows: a path $P_0$ between $(0,n-3)$ and $(0,n-2)$ in $G_0-\{(0,0),(0,n-1)\}$ with $2\le |V(P_0)|=m_1\le n-2$, a path $P_{(n-2)r-1}$ between $((n-2)r-1,1)$ and $((n-2)r-1,2)$ in $G_{(n-2)r-1}-\{((n-2)r-1,0),((n-2)r-1,n-1)\}$ with $2\le |V(P_{(n-2)r-1})|=m_2\le n-2$, a path $P_{(n-1)r}$ between $((n-1)r,1)$ and $((n-1)r,n-2)$ in $G_{(n-1)r}-\{((n-1)r,0),((n-1)r,n-1)\}$ with $2\le |V(P_{(n-1)r})|=m_3\le n-2$, a path $P_s^{\prime }$ between $(s,0)$ and $(s,n-1)$ in $G_s-(V(G_s)\cap \{e_1^{\prime },e_2^{\prime },e_3^{\prime }\})$ with $2\le |P_s^{\prime }|\le n-2$,where $s\in \{0,(n-2)r-1,(n-1)r\}$. Let

$$C_1=(C-\{e_1^{\prime },e_2^{\prime },e_3^{\prime }\})\cup P_0\cup P_{(n-2)r-1}\cup P_{(n-1)r}$$

and

$$C_2=(C^{\prime }-\{e_1^{″},e_2^{″},e_3^{″}\})\cup (\bigcup _{s\in \{0,(n-2)r-1,(n-1)r\}}{P}_s^{\prime }).$$

Then, $C_1$ and $C_2$ (see Figure 11a) are proved to be two disjoint cycles of $D(n,r)$, where $|V(C_1)|={𝓁}^{\prime }$, $|V(C_2)|=ng-{𝓁}^{\prime }$ and $V(C_1)\cup V(C_2)=V(D(n,r))$. Let ${𝓁}^{\prime }=6-|V({\cup }_{i=1,2,3}{e}_i^{\prime })|+{\sum }_{j=1}^3{m}_j$. Then $6\le {𝓁}^{\prime }\le 3n-6$.

Since $G_i\cong K_n$ for every i in the range $0\le i\le g-1$, we have the following paths: a path $Q_{(n-2)r-1}$ between $((n-2)r-1,1)$ and $((n-2)r-1,2)$ in $G_{(n-2)r-1}$ with $2\le |V(Q_{(n-2)r-1})|\le n$, a path $Q_{(n-1)r}$ between $((n-1)r,1)$ and $((n-1)r,n-2)$ in $G_{(n-1)r}$ with $2\le |V(Q_{(n-1)r})|\le n$. Also, we have the following edges: $g_1=(((n-2)r-2,0),((n-2)r,n-1))$ and $g_2=(((n-1)r-1,0),((n-1)r+1,n-1))$. Then, $C_1^{\prime }=(C_1\backslash P_{(n-2)r-1})\cup Q_{(n-2)r-1}$ and $C_2^{\prime }=(C_2\backslash P_{(n-2)r-1}^{\prime })\cup \left\{g_1\right\}$ are proved to be two disjoint cycles of $D(n,r)$, where $|V(C_1^{\prime })|={𝓁}^{\prime \prime }$, $|V(C_2^{\prime })|=ng-{𝓁}^{\prime \prime }$ and $V(C_1^{\prime })\cup V(C_2^{\prime })=V(D(n,r))$, with $n+4\le {𝓁}^{\prime \prime }\le 3n-4$, see Figure 11b.

Then $C_1^{\prime \prime }=(C_1^{\prime }-P_{(n-1)r})\cup Q_{(n-1)r}$ and $C_2^{\prime \prime }=(C_2^{\prime }-P_{(n-1)r})\cup \left\{g_2\right\}$ are proved to be two disjoint cycles of $D(n,r)$, where $|V(C_1^{\prime \prime })|={𝓁}^{\prime \prime \prime }$, $|V(C_2^{\prime \prime })|=ng-{𝓁}^{\prime \prime \prime }$ and $V(C_1^{\prime \prime })\cup V(C_2^{\prime \prime })=V(D(n,r))$, with $2n+2\le {𝓁}^{\prime \prime \prime }\le 3n-2$, see Figure 11c.

Since $n\ge 5$, then $n+4\le 3n-6$ and $2n+2\le 3n-3$. Thus, there exist two disjoint cycles, one of which has a length of ℓ and the other has a length of $ng-𝓁$ with $6\le 𝓁\le 3n-2$.

Case 3.

$3n-1\le 𝓁\le \lfloor {\displaystyle \frac{ng}{2}}\rfloor$.

We will show that $D(n,r)$ has two disjoint cycles $S_i$ and $S_i^{\prime }$, where $|V(S_i)|={𝓁}_i$, $|V(S_i^{\prime })|=ng-{𝓁}_i$ with $(i+1)n+4\le {𝓁}_i\le (i+3)n-2$, and $i=1,2,\cdots ,(n-3)r-2$.

By Definition 1 and $r\ge 2$, there exist two edges: $f=((0,\lfloor {\displaystyle \frac{(n-2)r-i-2}{r}}\rfloor ),((n-2)r-i-1,n-1-\lfloor {\displaystyle \frac{(n-2)r-i-2}{r}}\rfloor ))$ and $f^{\prime }=(((n-2)r,n-1-\lfloor {\displaystyle \frac{i+1}{r}}\rfloor )),((n-2)r-i-2,\lfloor {\displaystyle \frac{i+1}{r}}\rfloor ))$. Since $r\ge 2$ and $i\in \{1,2,\cdots ,(n-3)r-2\}$, then

$$1\le \lfloor {\displaystyle \frac{(n-2)r-i-2}{r}}\rfloor \le n-3.$$

In order to prove Case 3, we use the symbol in Case 2. Let $A=\{(n-2)r-i-1,(n-2)r-i,\cdots ,(n-2)r-2\}$ and $E^{\prime }={\cup }_{a\in A}\left\{((a,0),(a+1,n-1))\right\}$. Hamiltonian paths $T_a$ can be found between $(a,0)$ and $(a,n-1)$ in $G_a$ with $a\in A\backslash \{(n-2)r-i-1\}$. And we have a Hamiltonian path T between $((n-2)r-i-1,0)$ and $((n-2)r-i-1,n-1-\lfloor {\displaystyle \frac{(n-2)r-i-2}{r}}\rfloor )$ in $G_{(n-2)r-i-1}$ and a Hamiltonian path $T^{\prime }$ between $((n-1)r,0)$ and $((n-1)r,n-1)$ in $G_{(n-1)r}$.

So

$$S_i^1=(C_1^{\prime }-f_1)\cup \left\{f\right\}\cup T\bigcup _{a\in A}{T}_a\cup E^{\prime },$$

and

$$S_i^{1\prime }=(C_2\backslash \bigcup _{a\in A\cup \{(n-2)r-1\}}(V(G_a)))\cup \left\{f^{\prime }\right\}.$$

Then $S_i^1$ and $S_i^{1\prime }$ are two disjoint cycles, where $|V(S_i^1)|={𝓁}_i^1$, $|V(S_i^{1\prime })|=ng-{𝓁}_i^1$ with $(i+1)n+4\le {𝓁}_i^1\le (i+3)n-4$, see Figure 12a.

In addition, we have two disjoint cycles in $D(n,r)$: $S_i^2=(S_i^1-V(G_{(n-1)r}))\cup T^{\prime },$ and $S_i^{2\prime }=(S_i^{1\prime }\backslash V(G_{(n-1)r})\cup \left\{g_2\right\}$, where $|V(S_i^2)|={𝓁}_i^2$ and $|V(S_i^{2\prime })|=ng-{𝓁}_i^2$ with $(i+2)n+2\le {𝓁}_i^1\le (i+3)n-2$, see Figure 12b. Since $n\ge 5$, then $(i+3)n-3\ge (i+2)n+2$. Thus, there are two disjoint cycles $S_i$, $S_i^{\prime }$ of $D(n,r)$, where $|V(S_i)|={𝓁}_i$, $|V(S_i^{\prime })|=ng-{𝓁}_i$ with $(i+1)n+4\le {𝓁}_i\le (i+3)n-2$ and $i=1,2,\cdots ,(n-3)r-2$.

Since $n\ge 5$ and $i=1,2,\cdots ,(n-3)r-2$, $2n+4\le 3n-1$ and $(i+3)n-1\ge (i+2)n+4$. Also,

$$(i+3)n-2\le [(n-3)r-2+3]n-2=[(n-3)r+1]n-2.$$

So, $2n+4\le 𝓁\le [(n-3)r+1]n-2$.

If $n\ge 6$, then

$$\lfloor {\displaystyle \frac{ng}{2}}\rfloor -3rn-2=({\displaystyle \frac{n}{2}}-3)nr+{\displaystyle \frac{n}{2}}-2\ge 0,$$

that is, $\lfloor {\displaystyle \frac{ng}{2}}\rfloor \ge 3nr+2$. So $ng-\lfloor {\displaystyle \frac{ng}{2}}\rfloor \le ng-3nr-2$, that is, $\lfloor {\displaystyle \frac{ng}{2}}\rfloor \le [(n-3)r+1]n-2$.

Thus, if $n\ge 6$, $3n-1\le 𝓁\le \lfloor {\displaystyle \frac{ng}{2}}\rfloor$.

Now we show when $n=5$, Case 3 holds.

Since $n=5$ and $(i+1)n+4\le {𝓁}_i\le (i+3)n-2$ with $i\in \{1,2,\cdots ,(n-3)r-2\}$, then $𝓁\le 10r+3$. Now we will prove that two disjoint cycles R and $R^{\prime }$ can be found in $D(n,r)$, where $|V(R)|=𝓁$, $|V(R^{\prime })|=ng-𝓁$ with $10r+4\le 𝓁\le \lfloor {\displaystyle \frac{ng}{2}}\rfloor =\lfloor {\displaystyle \frac{5(5r+1)}{2}}\rfloor$.

By Definition 1 and $r\ge 2$, we have the following edges: $b=((0,1),(r+1,3))$, $b^{\prime }=((0,3),(3r+k,1))$, $b^{\prime \prime }=((r,2),(3r+k,2))$ with $k=1,2,\cdots ,r$. Let $B=\{r+2,r+3,\cdots ,3r+k-1\}$, $E^{\prime \prime }={\cup }_{x\in B}\left\{((x,0),(x+1,n-1))\right\}$. There exist Hamiltonian paths $T_x$ between $(x,0)$ and $(x,4)$ in $G_x$ with $x\in B$. And we have a Hamiltonian path W between $((r+1,0)$ and $((r+1,3)$ in $G_{r+1}$ and a Hamiltonian path $W^{\prime }$ between $(3r+k,4)$ and $(3r+k,1)$ in $G_{3r+k}$, a Hamiltonian path $W^{\prime \prime }$ between $(r,2)$ and $(r,0)$ in $G_r$.

Let $W_1^{\prime }$ be a path between $(0,0)$ and $(0,4)$ in $V(G_0)\backslash \{(0,1),(0,3)\}$, $W_1$ be a path between $(0,1)$ and $(0,3)$ in $V(G_0)\backslash \{(0,0),(0,4)\}$$W_2^{\prime }$ be a path between $(3r+k,0)$ and $(3r+k,2)$ in $G_{3r+k}\backslash \{(3r+k,1),(3r+k,4)\}$, $W_2$ be a path between $(3r+k,1)$ and $(3r+k,4)$ in $G_{3r+k}\backslash \{(3r+k,0),(3r+k,2)\}$, where $2\le |V(W_j)|\le 3$ and $2\le |V(W_j^{\prime })|\le 3$ with $j=1,2$.

There exist two disjoint cycles

$$R_1=\bigcup _{x\in B}{T}_x\cup \bigcup _{j=1,2}{W}_j\cup W\cup E^{\prime \prime }\cup \{b,b^{\prime }\}$$

and

$$R_1^{\prime }=(C_0\backslash (V(R_1)\cup V(G_r)))\cup \bigcup _{j=1,2}{W}_j\cup W^{\prime \prime }\cup \left\{b^{\prime \prime }\right\}.$$

Then $R_1$ and $R_1^{\prime }$ are two disjoint cycles of $D(n,r)$, where $|V(R_1)|=\alpha$, $|V(R_1^{\prime })|=ng-\alpha$ with $10r+4+(k-1)n\le \alpha \le 10r+6+(k-1)n$.

Also

$$R_2=(R_1-V(G_{3r+k}))\cup W^{\prime }$$

and

$$R_2^{\prime }=(R_1^{\prime }\backslash V(G_{3r+k})-\left\{b^{″}\right\})\cup \left\{((r,2),(3r+k+1,2))\right\}.$$

Then $R_2$ and $R_2^{\prime }$ are proved to be two disjoint cycles of $D(n,r)$, where $|V(R_1)|=\beta$, $|V(R_1^{\prime })|=ng-\beta$ with $10r+2+kn\le \beta \le 10r+3+kn$.

Since $k\in \{1,2,\cdots ,r\}$ and $r\ge 2$, so

$$15r+3\ge {\displaystyle \frac{5(5r+1)}{2}}.$$

And $10r+2+kn=10r+6+(k-1)n+1$.

Thus, there are two disjoint cycles R and $R^{\prime }$ of $D(n,r)$, where $|V(R)|=𝓁$, $|V(R^{\prime })|=ng-𝓁$ with $10r+4\le 𝓁\le \lfloor {\displaystyle \frac{ng}{2}}\rfloor =\lfloor {\displaystyle \frac{5(5r+1)}{2}}\rfloor$.

Considering all the cases discussed above, two vertex-disjoint cycles $C_1$ and $C_2$ can be found in $D(n,r)$, where $|V(C_1)|=𝓁$, $|V(C_2)|=ng-𝓁$ with $3\le 𝓁\le \lfloor {\displaystyle \frac{ng}{2}}\rfloor$. Therefore, $D(n,r)$ has two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋]$-pancyclicity for $n\ge 5$ and $r\ge 2$.

5. Conclusions and Future Work

This paper has successfully established the two-disjoint-cycle-cover pancyclicity of dragonfly networks. Specifically, we prove that $D(n,r)$ is two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋]$-pancyclic with $n\ge 3$ and $r\ge 2$, meaning that for any integer ℓ within $[3,⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋]$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ such that $|V(C_1)|=𝓁$ and $|V(C_2)|=|V(D(n,r))|-𝓁$. This result generalizes the known Hamiltonicity and vertex-pancyclicity properties of $D(n,r)$ demonstrated in prior work [2], thereby deepening the understanding of cycle embedding in interconnection networks. The two-disjoint-cycle-cover pancyclicity property is theoretically meaningful, as it reveals structural flexibility in dragonfly-like networks, which may enable more robust fault-tolerant routing and load-balancing mechanisms in high-performance computing systems.

The proof relies on a constructive method that exploits the symmetric topology of dragonfly networks, though it involves a case-based analysis to cover the range of cycle lengths. While this approach ensures rigor, future work could focus on developing more unified proof techniques, such as mathematical induction or parameterized constructions, to reduce the reliance on exhaustive cases and enhance insight. Additionally, algorithmic aspects, such as the efficient computation of these cycles in practical scenarios, also merit investigation.

Furthermore, a natural extension of this work is to investigate general graph properties that guarantee the existence of two-disjoint-cycle-cover pancyclicity, beyond the specific case of dragonfly networks. This aligns with the broader goal of deriving universal principles from cumulative analyses of special cases—including the many network topologies already studied (e.g., crossed cubes [3], augmented cubes [8,9]) and the dragonfly networks characterized here.

Additionally, dragonfly networks, which are of significant practical relevance in large-scale computing systems, have several underexplored fundamental properties that merit systematic investigation, such as three-path connectivity and fault-tolerant Hamiltonicity. Exploring these properties will deepen our theoretical understanding of dragonfly networks themselves and contribute incrementally to the empirical foundation needed for general graph-theoretic generalizations.