Strong Edge Coloring of K₄(t)-Minor Free Graphs

Abstract

A strong edge coloring of a graph G is a proper coloring of edges in G such that any two edges of distance at most 2 are colored with distinct colors. The strong chromatic index ${\chi }_s^{\prime }\left(G\right)$ is the smallest integer l such that G admits a strong edge coloring using l colors. A $K_4\left(t\right)$-minor free graph is a graph that does not contain $K_4\left(t\right)$ as a contraction subgraph, where $K_4\left(t\right)$ is obtained from a $K_4$ by subdividing edges exactly $t-4$ times. The paper shows that every $K_4\left(t\right)$-minor free graph with maximum degree $\mathsf{\Delta }\left(G\right)$ has ${\chi }_s^{\prime }\left(G\right)\le (t-1)\mathsf{\Delta }\left(G\right)$ for $t\in \{5,6,7\}$ which generalizes some known results on $K_4$-minor free graphs by Batenburg, Joannis de Verclos, Kang, Pirot in 2022 and Wang, Wang, and Wang in 2018. These upper bounds are sharp.

1. Introduction

In the whole paper, every considered graph is simple, which means an undirected graph without loops or multiple edges. The vertex set, the edge set and the maximum degree of a graph G are represented by $V\left(G\right)$, $E\left(G\right)$ and $\mathsf{\Delta }\left(G\right)$ (short for $\mathsf{\Delta }$), respectively. If the degree of a vertex v in G is k, then v is called a k-vertex. We call J a minor contained in G, denoted by $J⪯G$, if J is obtained from a subgraph of G by contracting edges. If G does not contain J as a minor, then G is called a J-minor free graph. Subdividing an edge $e=v_1{v}_2\in E\left(G\right)$ means removing edge e and adding a new vertex w with $w{v}_1,w{v}_2\in E\left(G\right)$. For a positive integer $t\ge 4$, define $K_4\left(t\right)$ to be a family of all isomorphic graphs obtained from $K_4$ by subdividing edges exactly $t-4$ times. From the definition above, it is clear that the collection of $K_4\left(t\right)$-minor free graphs is a subset of the collection of $K_4(t+1)$-minor free graphs.

For a graph G, let $d_G(e_1,e_2)$ denote the distance of two edges $e_1,e_2$ in G. If $e_1$ and $e_2$ are adjacent in G, then $d_G(e_1,e_2)=1$. If two non-adjacent edges $e_1$ and $e_2$ have a common adjacent edge in G, then $d_G(e_1,e_2)=2$. For an edge set $A\subseteq E\left(G\right)$ and a non-negative integer l, define $\varphi :A\to \{1,2,\cdots ,l\}$ to be a mapping satisfying $\varphi \left(e_1\right)\ne \varphi \left(e_2\right)$ whenever $d_G(e_1,e_2)\le 2$ for arbitrary two edges $e_1,e_2\in A$. Let $G\left[A\right]$ be the subgraph of G induced by edge set A. Such a mapping $\varphi$ above is called a partial strong edge coloring of $G\left[A\right]$. The smallest non-negative integer l such that $G\left[A\right]$ has a partial strong edge coloring is denoted by ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)$. A partial strong edge coloring of $G\left[A\right]$ is called a strong edge coloring of a graph G by taking $A=E\left(G\right)$. The smallest non-negative integer l for admitting a strong edge coloring of G with l colors is called the strong chromatic index of a graph G, denoted by ${\chi }_s^{\prime }\left(G\right)$.

Erdos and Nešetřil [1] proposed an outstanding conjecture for the strong chromatic index as follows.

Conjecture 1

(Erdos and Nešetřil, [1]). Let G be a graph with maximum degree Δ. Then

$${\chi }_s^{\prime }\left(G\right)\le \left\{\begin{array}{cc}1.25{\mathsf{\Delta }}^2,\hfill & \mathsf{\Delta } is even;\hfill \\ 1.25{\mathsf{\Delta }}^2-0.5\mathsf{\Delta }+0.25,\hfill & \mathsf{\Delta } is odd.\hfill \end{array}\right.$$

Andersen [2] proved that the strong chromatic index of cubic graphs is at most 10 for $\mathsf{\Delta }=3$; Cranston [3] obtained an upper bound 22 for $\mathsf{\Delta }=4$, which was further improved to 21 recently in [4] by Huang, Santatna and Yu. Mahdian [5] presented ${\chi }_s^{\prime }\left(G\right)\le (2+o\left(1\right))\frac{{\mathsf{\Delta }}^2}{ln\mathsf{\Delta }}$ for any $C_4$-free graph G, and Hurley, Joannis de Verclos, and Kang [6] recently proved that ${\chi }_s^{\prime }\left(G\right)\le 1.772{\mathsf{\Delta }}^2$ when $\mathsf{\Delta }$ is sufficently large.

This problem has also been intensively studied for many sparse graph classes. Wang [7] showed that ${\chi }_s^{\prime }\left(G\right)\le (4k-2)\mathsf{\Delta }-2k^2+1$ for any k-degenerate graph G with maximum degree $\mathsf{\Delta }$. When $k=2$, it was proved to ${\chi }_s^{\prime }\left(G\right)\le 5\mathsf{\Delta }+1$ by Choi, Kim, Kostochka, and Raspaud [8], and improved to $5\mathsf{\Delta }-{\mathsf{\Delta }}^{1/2-ϵ}+2$ when $\mathsf{\Delta }\ge 4^{1/\left(2ϵ\right)}$ for any $0<ϵ\le 1/2$ in [9]. Basavarjn and Francis [10] proved that ${\chi }_s^{\prime }\left(G\right)\le 3\mathsf{\Delta }$ for any chordless graph G with maximum degree $\mathsf{\Delta }$. Wang, Song, Wang, and Wang [11] showed that the strong chromatic index of any 1-planar graph is at most $14\mathsf{\Delta }$. More results can be found in [12,13,14,15].

Let $\mathcal{H}$ be a graph class. If for any $H\in \mathcal{H}$ and $J⪯H$ we have $J\in \mathcal{H}$, then this graph class $\mathcal{H}$ is called minor closed. In the studies of many graph properties, minor closed graph classes inherit some nice fundamental structures and have drawn extensive attention. As one of the most famous minor closed graph classes, the class of planar graphs is characterized by forbidding both $K_5$-minors and $K_{3,3}$-minors. Faudree, Gyárfás, Schelp, and Tuza [16] obtained the following upper bound of the strong chromatic index for planar graphs.

Theorem 1

(Faudree et al. [16]). For a planar graph G, ${\chi }_s^{\prime }\left(G\right)\le 4\mathsf{\Delta }+4$.

The proof of Theorem 1 is a combination of the famous Four Color Theorem (every planar graph is 4-colorable) and Vizing’s Theorem (every simple graph with maximum degree $\mathsf{\Delta }$ is $(\mathsf{\Delta }+1)$-edge colorable). We can obtain the following general theorem with a similar argument.

Theorem 2.

Let $\mathcal{H}$ be a minor closed graph class. Assume that, for some integer k, every graph $H\in \mathcal{H}$ is k-colorable. Then ${\chi }_s^{\prime }\left(H\right)\le k(\mathsf{\Delta }\left(H\right)+1)$ for any $H\in \mathcal{H}$.

Proof. 

By Vizing’s Theorem, we have an edge coloring of the graph H with $\mathsf{\Delta }\left(H\right)+1$ colors, where $E\left(G\right)$ is partitioned into $\mathsf{\Delta }\left(H\right)+1$ color classes, denoted by $E_1,E_2$, …, $E_{\mathsf{\Delta }+1}$. For $1\le i\le \mathsf{\Delta }+1$, obtain a graph $H_i=H/E_i$ from H by contracting the edges in $E_i$. Then by assumption, $H_i\in \mathcal{H}$ has a k-coloring ${\varphi }_i:E\left(H_i\right)\to \{1,2,\cdots ,k\}$. Note that, for any edge $e\in E_i$, its corresponding contracted vertex $x_e$ in $H_i$ receives a color ${\varphi }_i\left(x_e\right)$. Now, we define an edge coloring $\varphi :E\left(H\right)\to \{1,2,\cdots ,k(\mathsf{\Delta }+1\left)\right\}$ by setting $\varphi \left(e\right)=i{\varphi }_i\left(x_e\right)$ for each $e\in E_i$, where $1\le i\le \mathsf{\Delta }+1$. Then it is straightforward to check that $\varphi$ is a strong edge coloring of H, since any two edges of distance at most two receive distinct colors by the definition of $\varphi$.  □

In view of Theorem 2, the strong edge coloring problem is closely related to the following well-known Hadwiger conjecture proposed in 1943.

Conjecture 2

(Hadwiger, [17]). Every $K_t$-minor free graph is $(t-1)$-colorable.

Wagner [18] proved that the Four Color Theorem is equivalent to a slightly stronger statement that every $K_5$-minor-free graph is 4-colorable. Robertson, Seymour, and Thomas [19] showed that every $K_6$-minor-free graph is 5-colorable. Albar and Gonçalves [20] proved that any $K_7$-minor free graph is 8-colorable. Delcourt and Postle [21] showed that every $K_t$-minor free graph is $O(tloglogt)$-colorable, which is the best general result for Hadwiger conjecture currently. Considering Theorem 2, we immediately have results below.

Theorem 3.

If the Hadwiger conjecture is true, then every $K_t$-minor free graph has strong chromatic index at most $(t-1)(\mathsf{\Delta }+1)$. Moreover, each of the following statements holds.

(1)

When t is sufficiently large, every $K_t$-minor free graph G satisfies ${\chi }_s^{\prime }\left(G\right)\le \mathsf{\Delta }O(tloglogt)$.

(2)

When $t=7$, every $K_7$-minor free graph G satisfies ${\chi }_s^{\prime }\left(G\right)\le 8(\mathsf{\Delta }+1)$.

(3)

When $t=5,6$, every $K_6$-minor free graph G satisfies ${\chi }_s^{\prime }\left(G\right)\le 5(\mathsf{\Delta }+1)$, and every $K_5$-minor free graph G satisfies ${\chi }_s^{\prime }\left(G\right)\le 4(\mathsf{\Delta }+1)$.

However, improving the constant terms in Theorems 1 and 3 seems to be very difficult, as their proofs involve the usage of the Four Color Theorem. Note that Theorem 1 remains the best general upper bound for more than three decades, although this can be improved for some subclasses of planar graphs. For a smaller minor closed graph class, the class of $K_4$-minor free graphs, some improvements are known by Batenburg, Kang, Joannis de Verclos, Pirot [22] and Wang, Wang, and Wang [23] as follows.

Theorem 4

(Batenburg et al. [22]). Every $K_4$-minor free graph G with maximum degree Δ satisfies ${\chi }_s^{\prime }\left(G\right)\le 3\mathsf{\Delta }$.

Theorem 5

(Wang et al. [23]). For a $K_4$-minor free graph G with maximum degree $\mathsf{\Delta }\ge 3$, ${\chi }_s^{\prime }\left(G\right)\le 3\mathsf{\Delta }-2$.

In this study, we push further to get improvement for some wider graph classes, known as $K_4\left(t\right)$-minor free graphs for $t\in \{5,6,7\}$. In particular, the $t=7$ case of our results also provides some supporting evidence for the corresponding case of the Hadwiger conjecture.

Theorem 6.

For $t\in \{5,6,7\}$, if G is a $K_4\left(t\right)$-minor free graph with maximum degree Δ, then ${\chi }_s^{\prime }\left(G\right)\le (t-1)\mathsf{\Delta }$.

We actually prove a stronger result to obtain Theorem 6 as follows, where Theorem 6 is contained as a special case when $A=E\left(G\right)$.

Theorem 7.

If G is a $K_4\left(t\right)$-minor free graph where $t\in \{5,6,7\}$, then ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le (t-1)\mathsf{\Delta }\left(G\left[A\right]\right)$ for any edge set $A\subseteq E\left(G\right)$.

Note that, the upper bound in the theorem above is tight, where the graph G can be obtained from $K_{t-1}$ by adding some new vertices with degree 1 connecting each vertex of $K_{t-1}$.

In Section 2, we will first introduce the structure of 2-connected $K_4\left(t\right)$-minor free graphs for $t\in \{5,6,7\}$ characterized by Chen et al. in [24]. In Section 3, Theorem 7 will be proved for $t=5,6,7$ in Section 3.2, Section 3.3, Section 3.4 respectively, by the validity of a list lemmas presented in Section 3.1.

2. Decompositions of ${\mathit{K}}_{\mathbf{4}}\left(\mathit{t}\right)$-Minor Free Graphs

Recall that $K_4\left(t\right)$ is a family of all isomorphic graphs obtained from $K_4$ by subdividing edges exactly $t-4$ times. In this section, we introduce the equivalent conditions for 2-connected $K_4\left(t\right)$-minor free graphs proved by Chen, Fan, Lai, Song, and Xu [24], where $5\le t\le 7$. First, we introduce some notations.

Let $G_1,G_2$ be vertex disjoint simple graphs. We define two operations below.

(OP1) Suppose that $v_1,v_2\in V\left(G_1\right)$ and $v_1,v_2\in V\left(G_2\right)$. Let $G_1{\oplus }_{v_1,v_2}{G}_2$ be a simple graph gained from $G_1\cup G_2$ by identifying $v_1\in V\left(G_1\right)$ with $v_1\in V\left(G_2\right)$ to obtain a new vertex (again denoted by $v_1$) and $v_2\in V\left(G_1\right)$ with $v_2\in V\left(G_2\right)$ to obtain a new vertex (again denoted by $v_2$).

(OP2) Suppose that $e=xy\in E\left(G_1\right)$ and $t_1$-vertices $v_1,v_2\in V\left(K_{2,t_1}\right)$. Let $G_1{\oplus }_e{K}_{2,t_1}$ be a graph gained from $G_1\cup K_{2,t_1}$ by identifying x with $v_1$ to obtain a new vertex and y with $v_2$ to obtain a new vertex.

For a positive integer $t_1$, let $K_{2,t_1}=(X,Y)$ be a complete bipartite graph, where $X=\{v_1,v_2\}$ and $Y=\{u_1,u_2,\cdots ,u_{t_1}\}$. Define $K_{2,t_1}^{*}$ to be a graph gained from $K_{2,t_1}$ by adding one edge connecting $u_i$ and $u_{i+1}$ for each odd number i such that $1\le i\le t_1-1$. Let $\{v_1,v_2,\dots ,v_m\}$ be the vertex set of complete graph $K_m$. By operation (OP1), $K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4$ is defined and depicted in Figure 1a.

Figure 1. Graphs in (OP1) and (OP2).

Let $t_2$ and $g_1,g_2,\dots ,g_{t_2}$ be non-negative integers. Choose one edge either $v_1{w}_i$ or $v_2{w}_i$ in $K_{2,t_2}$ as $e_i$ for each i such that $1\le i\le t_2$. Let

$$T{P}_{2,t_2,(g_1,g_2,\dots ,g_{t_2})}=K_{2,t_2}{\oplus }_{e_1}{K}_{2,g_1}{\oplus }_{e_2}{K}_{2,g_2}\dots {\oplus }_{e_{t_2}}{K}_{2,g_{t_2}},$$

be a graph gained from $K_{2,t_2}$ and $K_{2,g_i}$ by successively applying operation (OP2) once for each $e_i$ such that $1\le i\le t_2$. The graph $T{P}_{2,t_2,(g_1,g_2,\dots ,g_{t_2})}$ is depicted in Figure 1b.

Let $\mathcal{J}$ be a collection of subgraphs of H and $[J,H]=\{K:forsomeJ\in \mathcal{J},V(J)\subseteq V(K)\subseteq V(H),E(J)\subseteq E(K)\subseteq E(H\left)\right\}$. Chen et al. proved the following results in [24].

Proposition 1

(Chen et al. [24]). Every 2-connected graph G is $K_4\left(5\right)$-minor free if and only if G is a $K_4$-minor free graph or $K_4$.

Proposition 2

(Chen et al. [24]). Every 2-connected graph G is $K_4\left(6\right)$-minor free if and only if G is contained in the set $[K_4\left(5\right),K_5]$ or G is a $K_4\left(5\right)$-minor free graph or G is contained in the set $\mathcal{Q}={\cup }_{t_1\ge 1}\{K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4-v_1{v}_2,K_{2,t_1}{\oplus }_{v_1,v_2}-K_4\}$.

Let $Q_1=K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4{\oplus }_{v_3{v}_{\ell }}{K}_{2,t_2}$ be a graph obtained from $K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4$ and $K_{2,t_2}$ by applying operation (OP2) once for the edge $e=v_3{v}_{\ell }$ in $K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4$, where $\ell \in \{1,4\}$. When $\ell =1$, the graph $Q_1$ is depicted in Figure 1c. Let $Q_2=K_{2,t_1}^{*}{\oplus }_{v_1,v_2}{K}_4{\oplus }_{v_1,v_2}T{P}_{2,t_2,(g_1,g_2,\dots ,g_{t_2})}$ be a graph gained from $K_{2,t_1}^{*}{\oplus }_{v_1,v_2}{K}_4$ and $T{P}_{2,t_2,(g_1,g_2,\dots ,g_{t_2})}$ by applying operation (OP1) once on vertices $v_1,v_2$. The graph $Q_2$ is depicted in Figure 2.

Figure 2. The graph $Q_2$.

Proposition 3

(Chen et al. [24]). Every 2-connected graph G is $K_4\left(7\right)$-minor free if and only if G is in the set $[K_4\left(6\right),K_6]$ or G is a $K_4\left(6\right)$-minor free graph or in the set ${\mathcal{Q}}_i$ where

(1)

${\mathcal{Q}}_1=\{G\in [M_1,Q_1]:|V\left(G\right)|\ge 6\}$, where $M_1=Q_1-\{v_1{v}_2,v_3{v}_{\ell }\}$ and $\ell \in \{1,4\}$.

(2)

${\mathcal{Q}}_2=\{G\in [M_2,Q_2]:|V\left(G\right)|\ge 6\}$, where $M_2=Q_2-v_1{v}_2-{\bigcup }_{i=1}^{t_2}{e}_i.$

(3)

${\mathcal{Q}}_3=\{G\in \left[M_3,Q_3\right]:\left|V\left(G\right)\right|\ge 6\}$, where $Q_3=K_{t_1,2}{\oplus }_{v_1,v_2}{K}_5,M_3=Q_3-\{v_1{v}_2,v_1{v}_3,v_2{v}_5\}$.

3. Strong Edge Coloring

In this section, we prove our main result Theorem 7. For a graph G, define a core $C\left(G\right)$ to be a graph gained from G by internally deleting 1-vertices so that $\delta \left(C\right(G\left)\right)\ge 2$. We first consider block decomposition of $C\left(G\right)$ and give general lemmas for every $K_4\left(t\right)$-minor free graph whenever $5\le t\le 7$ in Section 3.1. Next, we prove Theorem 7 when $t=5,6,7$ in Section 3.2, Section 3.3 and Section 3.4, respectively.

Take G as a minimal counterexample for Theorem 7 with $\left|V\right(C\left(G\right)\left)\right|$ minimized and subject to this condition we choose $\left|E\right(G\left)\right|$ minimized. Choose an arbitrary edge set $A\subseteq E\left(G\right).$

3.1. Block Decomposition of $C\left(G\right)$ and General Lemmas

A leaf block in a graph G is a maximal 2-connected subgraph of G containing exactly one cut vertex of G. Let $B_1$ be a leaf block of $C\left(G\right)$. We use $a\left(B_1\right)$ to denote the cut vertex of $C\left(G\right)$ connecting $B_1$ and the remaining part in $C\left(G\right)$. We call e an interior edge in $B_1$ if e is not incident with $a\left(B_1\right)$. We call v an interior vertex in $B_1$ if $v\ne a\left(B_1\right)$. For any $x\in V\left(C\right(G\left)\right)$, let $T_x$ denote the tree generated by the vertices and edges deleted from G starting from vertex x so that $C\left(G\right)$ is obtained.

For $e\in A$, let $N\left(e\right)=\{e^{\prime }:e^{\prime }\in A$ and $d_G(e,e^{\prime })\le 2\}$ and $\left|N\right(e\left)\right|$ be the order of $N\left(e\right)$. For $e=xy\in E\left(G\right)$, let $E_{G^{\prime }}\left(x\right)$ denote the edges incident with x in $G^{\prime }$, where $G^{\prime }=G-e$. Denote $d_{G^{\prime }}(x,y)=max\{d_{G^{\prime }}(e_1,e_2):$$\forall e_1\in E_{G^{\prime }}\left(x\right)$ and $\forall e_2\in E_{G^{\prime }}\left(y\right)\}$. Especially, let $d_{G^{\prime }}(x,y)=0$ if either $d_G\left(x\right)=1$ or $d_G\left(y\right)=1$.

Lemma 1.

Let $e=xy$ be an edge in $E\left(G\right)$. Then at least one of the following conditions does not hold.

(1)

$d_{G^{\prime }}(x,y)\le 2$;

(2)

$\left|N\right(e\left)\right|\le (t-1)\mathsf{\Delta }\left(G\right[A\left]\right)-1$ for $t\in \{5,6,7\}$.

Proof. 

By contradiction, assume that we have one edge $e\in E\left(B_1\right)$ satisfying conditions $\left(1\right)$ and $\left(2\right)$. Let $G^{\prime }=G-e$, where $e=xy$. Thus ${\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)$ for any $A_0\subseteq E\left(G^{\prime }\right)$ as G is minimized. We choose an arbitrary $A\subseteq E\left(G\right)$. If $e\notin A$, then we take $A_0=A$, and so ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)={\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)=(t-1)\mathsf{\Delta }\left(G\left[A\right]\right)$ by $G\left[A\right]=G^{\prime }\left[A_0\right]$, contrary to the assumption of G. Next, we assume $e\in A$ and let $A_0=A\setminus e$. Now we have $G\left[A\right]=G^{\prime }\left[A_0\right]\bigcup \left\{e\right\}$ and $\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)\le \mathsf{\Delta }\left(G\left[A\right]\right)$. Let $\varphi$ be a partial strong edge coloring of $G^{\prime }\left[A_0\right]$ with $(t-1)\mathsf{\Delta }\left(G\right[A\left]\right)$ colors. Since $d_{G^{\prime }}(x,y)\le 2$, any two edges $e_1,e_2$ in A satisfying $e_1\in E_{G^{\prime }}\left(x\right)$ and $e_2\in E_{G^{\prime }}\left(y\right)$ are colored distinct in coloring $\varphi$. Especially, when $d_{G^{\prime }}(x,y)=0$, either $E_{G^{\prime }}\left(x\right)\bigcap A=\varnothing$ or $E_{G^{\prime }}\left(y\right)\bigcap A=\varnothing$. By $\left|N\right(e\left)\right|\le (t-1)\mathsf{\Delta }\left(G\right[A\left]\right)-1$, we would choose one available color for edge e. Thus, $G\left[A\right]$ has a partial strong edge coloring with $(t-1)\mathsf{\Delta }\left(G\right[A\left]\right)$ colors which implies that ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le (t-1)\mathsf{\Delta }\left(G\left[A\right]\right)$, a contradiction. This proves the lemma.  □

Lemma 2.

Let x be an interior vertex in $B_1$. Then $T_x$ is a star graph.

Proof. 

Arguing by contradiction, assume that $T_x$ is not a star graph. Pick an edge $e\in E\left(T_x\right)$ such that $e=v_1{v}_2$ is the outermost edge of $T_x$ with $d_G\left(v_1\right)=1$. Delete the edge e and let $G^{\prime }=G-e$. By $d_G\left(v_1\right)=1$, we obtain $d_{G^{\prime }}(v_1,v_2)=0$ and $\left|N\right(e\left)\right|\le 2\mathsf{\Delta }\left(G\right[A\left]\right)$, a contradiction to Lemma 1.  □

Lemma 3.

Let v be an interior vertex in $B_1$ with $d_{C\left(G\right)}\left(v\right)\le t-3$ for $t\in \{5,6,7\}$. Then $d_{C\left(G\right)}\left(v\right)=d_G\left(v\right)$.

Proof. 

By contradiction, assume $d_{C\left(G\right)}\left(v\right)\ne d_G\left(v\right)$ which implies $T_v\ne \varnothing$. By Lemma 2, we have $T_v$ is a star graph and choose $e=xv\in E\left(T_v\right)$. Delete the edge e and let $G^{\prime }=G-e$. By $d_G\left(x\right)=1$, we obtain $d_{G^{\prime }}(x,y)=0$ and $\left|N\right(e\left)\right|\le (t-2)\mathsf{\Delta }\left(G\right[A\left]\right)$, which is a contradiction to Lemma 1.  □

Lemma 4.

Let v be an interior vertex contained in $B_1$ with $|B_1|\le t-1$, where $t\in \{5,6,7\}$. Then $d_G\left(v\right)=d_{C\left(G\right)}\left(v\right)$.

Proof. 

By contradiction, assume $d_{C\left(G\right)}\left(v\right)\ne d_G\left(v\right)$ which implies $T_v\ne \varnothing$. By Lemma 2, we have $T_v$ as a star graph and pick an edge $e=xv\in E\left(T_v\right)$. Let $G^{\prime }=G-\left\{e\right\}$. By the minimized counterexample property, ${\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)$ for any $A_0\subseteq E\left(G^{\prime }\right)$.

Choose an arbitrary edge set $A\subseteq E\left(G\right)$. When $e\notin A$, choose $A_0=A$, and then $G\left[A\right]=G^{\prime }\left[A_0\right]$. Thus, ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)={\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)=(t-1)\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$, a contradiction. Otherwise $e\in A$, let $A_0=A-\left\{e\right\}$ and we have $G^{\prime }\left[A_0\right]\cup \left\{e\right\}=G\left[A\right]$. Thus, we obtain $\left|N\right(e\left)\right|\le (t-1)\mathsf{\Delta }\left(G\right[A\left]\right)-1$ and $d_{G^{\prime }}(x,v)=0$ by $d_G\left(x\right)=1$, which is a contradiction to Lemma 1.  □

Lemma 5.

For an interior edge $e=v_1{v}_2$ contained in $B_1$ with $|B_1|\le t-1$, we have $d_{G^{\prime }}(v_1,v_2)>2$, where $G^{\prime }=G-\left\{e\right\}$ and $t\in \{5,6,7\}$.

Proof. 

By contradiction, assume that there exists an interior edge $e=v_1{v}_2$ such that $d_{G^{\prime }}(v_1,v_2)\le 2$. Let $G^{\prime }=G-\left\{e\right\}$. By the minimized counterexample property, ${\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)$ for any set $A_0\subseteq E\left(G^{\prime }\right)$.

Pick an arbitrary edge set $A\subseteq E\left(G\right)$. When $e\notin A$, take $A_0=A$, and then ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)={\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)=(t-1)\mathsf{\Delta }\left(G\left[A\right]\right)$ by $G\left[A\right]=G^{\prime }\left[A_0\right]$ for any $A\subseteq E\left(G\right)$. If $e\in A$, let $A_0=A-e$ and we have $G^{\prime }\left[A_0\right]\cup \left\{e\right\}=G\left[A\right]$. Thus we obtain $\left|N\right(e\left)\right|\le (t-1)\mathsf{\Delta }\left(G\right[A\left]\right)-1$ by Lemma 4 and $|B_1|\le t-1.$ It is a contradiction to $d_{G^{\prime }}(v_1,v_2)\le 2$ by Lemma 1.  □

Corollary 1.

Every leaf block $B_1$ is not isomorphic to $K_{t-1}$ for $t\in \{5,6,7\}$.

Proof. 

Suppose $B_1\cong K_{t-1}$ and choose an interior edge $e=v_1{v}_2\in E\left(K_{t-1}\right)$. Let $G^{\prime }=G-e$. We have $d_{G^{\prime }}(v_1,v_2)\le 2$ by the definition of $d_{G^{\prime }}(v_1,v_2)$, which is a contradiction to Lemma 5.  □

Lemma 6.

For any interior vertex x contained in $B_1$, $d_{B_1}\left(x\right)\ne 2$.

Proof. 

By contradiction, suppose that there exists an interior vertex x with $d_{C\left(G\right)}\left(x\right)=d_{B_1}\left(x\right)=2$. By Lemma 3, we have $d_G\left(x\right)=d_{C\left(G\right)}\left(x\right)=2$ and the vertex x has exactly two neighbours $y_1,y_2$. We characterize the graph $G^{\prime }$ obtained from G by deleting x and adding $x_1$ and $x_2$ as follows: adding a 1-vertex $x_i$ and an edge $x_i{y}_i$ for $i\in \{1,2\}$; adding an edge $y_1{y}_2$ if $y_1{y}_2$ does not exist in $B_1$.

Consider the core of $G^{\prime }$, represented by $C\left(G^{\prime }\right)$. As $|V\left(C\left(G^{\prime }\right)\right)|=|V\left(C\left(G\right)\right)|-1$, by the minimality of counterexample, ${\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)$ for arbitrary $A_0\subseteq E\left(G^{\prime }\right)$. To show ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le (t-1)\mathsf{\Delta }\left(G\left[A\right]\right)$ for arbitrary $A\subseteq E\left(G\right)$, let $A^{\prime }\subseteq E\left(G^{\prime }\right)$ be the set gained from $A\subseteq E\left(G\right)$ by replacing $x{y}_i$ with $x_i{y}_i$, if $x{y}_i$ exists in A for $i\in \{1,2\}$. Let $\varphi$ be a desired coloring of $G^{\prime }\left[A{}^{\prime }\right]$ with $(t-1)\mathsf{\Delta }\left(G^{\prime }\left[A^{\prime }\right]\right)$ colors.

If $y_1{y}_2\in E\left(G\right)$, then it implies $y_1{y}_2\in E\left(G^{\prime }\right)$. If $\{x{y}_1,x{y}_2\}\subseteq A$, let $A^{\prime }=A\bigcup \{x_1{y}_1,x_2{y}_2\}$$\setminus \{x{y}_1,x{y}_2\}$ and then $\mathsf{\Delta }\left(G^{\prime }\left[A^{\prime }\right]\right)\le \mathsf{\Delta }\left(G\left[A\right]\right).$ If either $x{y}_1\notin A$ or $x{y}_2\notin A$, then $A^{\prime }\cong A$ and $\mathsf{\Delta }\left(G^{\prime }\left[A^{\prime }\right]\right)=\mathsf{\Delta }\left(G\left[A\right]\right).$ Let $\phi$ be a new coloring of A defined as follows.

$$\phi \left(e\right)=\left\{\begin{array}{cc}\varphi \left(x_1{y}_1\right),\hfill & e=x{y}_1;\hfill \\ \varphi \left(x_2{y}_2\right),\hfill & e=x{y}_2;\hfill \\ \varphi \left(e\right),\hfill & e\in A-\left\{x{y}_1,x{y}_2\right\}.\hfill \end{array}\right.$$

Thus $\phi$ is a desired coloring of $G\left[A\right]$ with ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A^{\prime }\right]\right)\le (t-1)\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$, a contradiction to the assumption of G.

If $y_1{y}_2\notin E\left(G\right),$ then $y_1{y}_2\in E\left(G^{\prime }\right)$ by definition. Note that $y_1{y}_2\notin A$ and $y_1{y}_2\notin A^{\prime }$. If $\{x{y}_1,x{y}_2\}\subseteq A$, then $A^{\prime }=A\bigcup \{x_1{y}_1,x_2{y}_2\}\setminus \{x{y}_1,x{y}_2\}$ and $\mathsf{\Delta }\left(G^{\prime }\left[A^{\prime }\right]\right)\le \mathsf{\Delta }\left(G\left[A\right]\right).$ If either $x{y}_1\notin A$ or $x{y}_2\notin A$, then $A^{\prime }\cong A$ and $\mathsf{\Delta }\left(G^{\prime }\left[A^{\prime }\right]\right)=\mathsf{\Delta }\left(G\left[A\right]\right)$. In each case, $\mathsf{\Delta }\left(G^{\prime }\left[A^{\prime }\right]\right)\le \mathsf{\Delta }\left(G\left[A\right]\right)$ is obtained. Let $\phi$ be a coloring of $G\left[A\right]$ defined as below.

$$\phi \left(e\right)=\left\{\begin{array}{cc}\varphi \left(x_1{y}_1\right),\hfill & e=x{y}_1;\hfill \\ \varphi \left(x_2{y}_2\right),\hfill & e=x{y}_2;\hfill \\ \varphi \left(e\right),\hfill & e\in A-\left\{x{y}_1,x{y}_2\right\}.\hfill \end{array}\right.$$

Thus $\phi$ is a desired coloring of $G\left[A\right]$ and ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le (t-1)\mathsf{\Delta }\left(G^{\prime }\left[A^{\prime }\right]\right)\le (t-1)\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$, which is a contradiction.  □

Note that all lemmas above in Section 3.1 are valid for the minimal counterexample G whenever $5\le t\le 7$. Next we apply these lemmas and characterize graph structures to prove Theorem 7 for each $5\le t\le 7$ below in Section 3.2, Section 3.3 and Section 3.4, respectively.

3.2. Proof of Strong Edge Coloring of $K_4\left(5\right)$-Minor Free Graphs

To get a useful result below, we need to introduce the definition of 2-tree. A 2-tree refers to a graph obtained from $K_3$ by sequentially adding a new vertex and connecting it to two adjacent vertices. It is well-known that $K_4$-minor free graphs are exactly subgraphs of 2-trees.

Lemma 7.

For any 2-connected $K_4$-minor free graph G with $\left|V\right(G\left)\right|\ge 4$, there exist at least two non-adjacent 2-vertices in G.

Proof. 

Since G is a subgraph of some 2-tree H, it is enough to prove this statement for all 2-trees. Note that $2\le d_G\left(v\right)\le d_H\left(v\right)$ for arbitrary vertex v in G. Now we are to prove that any 2-tree H with $\left|V\right(H\left)\right|\ge 4$ contains at least two non-adjacent 2-vertices by induction on $\left|V\right(H\left)\right|$. By definition of 2-tree, the conclusion is clear when $\left|V\right(H\left)\right|=4$. Assume that the conclusion is true when $\left|V\right(H\left)\right|\le p$. When $\left|V\right(H\left)\right|=p+1$, by the construction of a 2-tree, H can be obtained from a 2-tree $H^{\prime }$ with $\left|V\right(H^{\prime }\left)\right|=p$ by adding one new 2-vertex v and connecting v with two distinct vertices $v_1,v_2$ in $H^{\prime }$ where $v_1{v}_2\in E\left(H^{\prime }\right)$. Since H destroys at most one 2-vertex contained in $H^{\prime }$ and one new 2-vertex v is added to H, we conclude that H contains at least two non-adjacent 2-vertices as required.  □

For a $K_4\left(5\right)$-minor free graph G, we need to prove ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 4\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$. By Proposition 1 and Corollary 1, $B_1$ is a 2-connected $K_4$-minor free graph. By Lemma 7, $B_1$ contains at least one 2-vertex as interior vertex, which is a contradiction to Lemma 6. Thus we have ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 4\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$.

3.3. Proof of Strong Edge Coloring of $K_4\left(6\right)$-Minor Free Graphs

For a $K_4\left(6\right)$-minor free graph G, we need to prove ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 5\mathsf{\Delta }\left(G\left[A\right]\right)$ for any edge set $A\subseteq E\left(G\right)$. First, we give the following useful lemma for a $K_4\left(6\right)$-minor free graph $G.$

Lemma 8.

Let $e=v_1{v}_2$ be an interior edge contained in $B_1$ with $|B_1|=5$. Let $G^{\prime }=G-\left\{e\right\}$. For any $u_i$ as a neighbour of $v_i$ where $i\in \{1,2\}$, if either $d_{C\left(G\right)}\left(u_1\right)\ge 3$ or $d_{C\left(G\right)}\left(u_2\right)\ge 3$ holds, then $d_{G^{\prime }}(v_1,v_2)\le 2.$

Proof. 

Without loss of generality, assume that $d_{C\left(G\right)}\left(u_1\right)\ge 3.$ Since $|B_1|=5$ and $d_{C\left(G\right)}\left(u_1\right)\ge 3$, we have either $u_1{u}_2\in E\left(G\right)$ or $u_1{v}_2\in E\left(G\right)$ and $d_{G^{\prime }}(v_1{u}_1,v_2{u}_2)\le d_{C\left(G\right)}(v_1{u}_1,v_2{u}_2)\le 2$. By the arbitrariness of $u_1,u_2$, $d_{G^{\prime }}(v_1,v_2)\le d_{C\left(G\right)}(v_1{u}_1,v_2{u}_2)\le 2$.  □

By Proposition 3, we get $B_1\in [K_4\left(5\right),K_5]$ or $B_1$ is a $K_4\left(5\right)$-minor free graph or $B_1\in \mathcal{Q}={\cup }_{t_1\ge 1}\{K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4-v_1{v}_2,K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4\}$. We analyze the leaf block $B_1$ case by case.

Case 1. $B_1\in [K_4\left(5\right),K_5]$.

The class of graphs $[K_4\left(5\right),K_5]$ can be characterized in $\{H_1,H_2,H_3,H_4,H_5\}$ (see Figure 3). For $B_1\in \{H_1,H_2\}$, we obtain $a\left(B_1\right)=w\in H_i$ for $i\in \{1,2\}$ by Lemma 6. Choose an interior edge $e=v_1{v}_2\in E\left(H_i\right)$ and let $G^{\prime }=G-\left\{e\right\}.$ By simple computation, we have $d_{G^{\prime }}(v_1,v_2)\le 2$, which is a contradiction to Lemma 5.

Figure 3. All graphs of $[K_4\left(5\right),K_5]$.

By Corollary 1, we have $B_1\in \{H_3,H_4\}$. Take an interior edge $e=v_1{v}_2\in E\left(B_1\right)$ and let $G^{\prime }=G-\left\{e\right\}$. As $d_{C\left(G\right)}\left(v\right)\ge 3$ for any $v\in V\left(B_1\right)$, $d_{G^{\prime }}(v_1,v_2)\le 2$ is obtained by Lemma 8. By Lemma 5, we have ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 5\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right).$

Case 2. $B_1\in {\cup }_{t_1\ge 1}\{K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4-v_1{v}_2,K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4\}$.

If $B_1=K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4$, then we gain $t_1=1$, $a\left(B_1\right)=u_1$ and $B_1=H_2$ by Lemma 6. If $B_1=K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4-v_1{v}_2$, then we obtain $t_1=1$, $a\left(B_1\right)=u_1$ and $B_1=H_1$ by Lemma 6. In these two conditions, by Case 1, the desired result is achieved.

Case 3. $B_1$ is a 2-connected $K_4\left(5\right)$-minor free graph.

By Proposition 1 and Corollary 1, $B_1$ is a $K_4$-minor free graph. By Lemma 7, $B_1$ contains at least one 2-vertex as interior vertex, contradicting Lemma 6.

3.4. Proof of Strong Edge Coloring of $K_4\left(7\right)$-Minor Free Graphs

For a $K_4\left(7\right)$-minor free graph G, we need to prove ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 6\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$. By Proposition 3, we obtain $B_1\in [K_4\left(6\right),K_6]$ or $B_1\in {\mathcal{Q}}_i$ ($i\in \{1,2,3\}$) or $B_1$ is a $K_4\left(6\right)$-minor free graph. We prove the statement based on the following three cases.

Case 1. $B_1\in [K_4\left(6\right),K_6]$.

For any $v\in V\left(B_1\right)\setminus \left\{a\left(B_1\right)\right\}$, we have $d_{C\left(G\right)}\left(v\right)\ge 3$ by Lemma 6. Let $e=v_1{v}_2$ be an arbitrary interior edge contained in leaf block $B_1\in [K_4\left(6\right),K_6]$ and let $G^{\prime }=G-\left\{e\right\}$. By Lemma 5, $d_{G^{\prime }}(v_1,v_2)\le 2$ does not hold. Thus there exist $e_1=v_1{u}_1\in E\left(C\left(G\right)\right)$ and $e_2=u_2{v}_2\in E\left(C\left(G\right)\right)$ satisfying $d_{G^{\prime }}(e_1,e_2)>2.$ By simple computation, we enumerate the following results.

• Fact A:

(1)

$e\notin E\left(C\right(G\left)\right)$, when $e\in \{v_1{u}_2,u_1{v}_2,u_1{u}_2\}$.

(2)

$d_{C\left(G\right)}\left(u_1\right)\le 3$ and $d_{C\left(G\right)}\left(u_2\right)\le 3$.

(3)

$d_{C\left(G\right)}\left(v\right)\ge 3$ for any $v\in V\left(B_1\right)\setminus \left\{a\left(B_1\right)\right\}.$

By symmetry and Lemma 6, we have either $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,3\}$ or $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,2\}$ for Fact A(2). If $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,3\}$ and $zw\notin E\left(C\right(G\left)\right)$, then we characterize all possible graphs $N_i$ for $1\le i\le 6$ in Figure 4. If $B_1=N_3$, then $a\left(B_1\right)=w$ and take $e=v_{2}z$. Let $G^{\prime }=G-\left\{e\right\}$ and we have $d_{G^{\prime }}(v_2,z)\le 2$, which is a contraction to Lemma 5. If $B_1\in \{N_1,N_2,N_4,N_5,N_6\}$, then choose a thick edge $e\in \{v_1{u}_1,v_2{u}_2\}\subseteq E\left(N_i\right)$ marked in Figure 4 such that e is an interior edge contained in $B_1$. Let $G^{\prime }=G-\left\{e\right\}$ and we have $d_{G^{\prime }}(v_i,u_i)\le 2$ by simple computation for some $i\in \{1,2\}$, which is a contraction to Lemma 5.

Figure 4. All graphs satisfied $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,3\}$ and $zw\notin E\left(G\right)$.

Let $N_i^{\prime }=N_i\cup \left\{zw\right\}$ for $1\le i\le 6$. If $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,3\}$ and $zw\in E\left(C\right(G\left)\right)$, then $B_1\cong N_i^{\prime }$ for some $1\le i\le 6$. If either $B_1\cong N_i^{\prime }$ where $i\in \{1,2,4,5,6\}$ or $B_1\cong N_3^{\prime }$ (see Figure 5) and $a\left(B_1\right)\ne z$, then we can do the same analysis as $N_i$ where $1\le i\le 6$. If $B_1\cong N_3^{\prime }$ and $a\left(B_1\right)=z$, then we can first pick a cycle $C=u_1{e}_1{v}_1{e}_2{v}_2{e}_3{u}_2{e}_{4}w{e}_5{u}_1$. Then delete C and let $G^{\prime }=G\setminus E\left(C\right)$, by the minimized counterexample property, ${\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le 6\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)$ for arbitrary $A_0\subseteq E\left(G^{\prime }\right)$. Choose an arbitrary $A\subseteq E\left(G\right)$, then we consider the following two subcases.

Figure 5. The graph $N_3^{\prime }$.

Subcase 1.1 $e_i\notin A$ for $1\le i\le 5$.

Let $A_0=A$. Then $G\left[A\right]=G^{\prime }\left[A_0\right]$. We can get ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)={\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le 6\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)$$=6\mathsf{\Delta }\left(G\right[A\left]\right)$ for any $A\subseteq E\left(G\right)$, a contradiction.

Subcase 1.2 $A^{*}=A\cap \{e_1,e_2,e_3,e_4,e_5\}$ and $A^{*}\ne \varnothing$.

Let $A_0=A\setminus A^{*}$. Then $\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)\le \mathsf{\Delta }\left(G\left[A\right]\right)$. Take $\varphi$ as a partial strong edge coloring of graph $G^{\prime }\left[A_0\right]$ with $6\mathsf{\Delta }\left(G\right[A\left]\right)-1$ colors. Choose $|A^{*}|$ different colors from $6\mathsf{\Delta }\left(G\right[A\left]\right)-1$ colors to assign distinct edges in $A^{*}$ so that those colors are different from any $\varphi \left(e\right)$, where $e\in A\bigcap E_G\left(z\right)$. Therefore, we gain an desired coloring of $G\left[A\right]$ with ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le max\{\mathsf{\Delta }\left(G\left[A\right]\right)+5,6\mathsf{\Delta }\left(G\left[A\right]\right)-1\}\le 6\mathsf{\Delta }\left(G\left[A\right]\right)-1$ for any $A\subseteq E\left(G\right)$, a contradiction.

If $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,2\}$ and $zw\notin E\left(C\right(G\left)\right)$, then we have $d_{C\left(G\right)}\left(v_i\right)\ge 4$ for some $i\in \{1,2\}$ by Lemma 6. We characterize all possible graphs $T_i$ for $1\le i\le 6$ in Figure 6 corresponding to $\{d_{C\left(G\right)}\left(v_1\right),d_{C\left(G\right)}\left(v_2\right)\}\in \{\{3,4\},\{4,3\},\{4,4\}\}$ by Fact A. If $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,2\}$ and $zw\in E\left(C\right(G\left)\right)$, then we have all possible graphs $P_i$ for $1\le i\le 6$ in the Figure 7 when $\{d_{C\left(G\right)}\left(v_1\right),d_{C\left(G\right)}\left(v_2\right)\}=\{3,3\}$.

Figure 6. All graphs satisfied $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,2\}$ and $zw\notin E\left(G\right)$.

Figure 7. All graphs satisfied $\{d_{C\left(G\right)}\left(v_1\right),d_{C\left(G\right)}\left(v_2\right),d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,3,3,2\}$ and $zw\in E\left(G\right)$.

For each of the graphs in Figure 6 and Figure 7, we know that $a\left(B_1\right)=u_2$ in each graph by Lemma 6. We would gain a new graph $G^{\prime }$ by deleting the thick edge $e=xy$. We have $d_{G^{\prime }}(x,y)\le 2$ by simple computation, which is a contradiction to Lemma 5. When $\{d_{C\left(G\right)}\left(u_1\right),d_{C\left(G\right)}\left(u_2\right)\}=\{3,2\}$, $zw\in E\left(G\right)$ and $\{d_{C\left(G\right)}\left(v_1\right),d_{C\left(G\right)}\left(v_2\right)\}\in \{\{3,4\},\{4,3\},$$\{4,4\}\}$, we get the desired result as $P_i$ since $B_1$ contains $P_i$ as a subgraph for some $1\le i\le 6$.

Case 2. $B_1\in \{G\in [M_1,Q_1]:|V\left(G\right)| \ge 6\}$.

Recall that $Q_1=K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4{\oplus }_{v_3{v}_{\ell }}{K}_{2,t_2}$ and $M_1=Q_1-\{v_1{v}_2,v_3{v}_{\ell }\}$. By construction of $Q_1$ and $\left|V\right(B_1\left)\right|\ge 6$, there is at least one 2-vertex as interior vertex in $B_1$, which is a contradiction to Lemma 6.

Case 3. $B_1\in \{G\in [M_2,Q_2]:|V\left(G\right)| \ge 6\}$.

Recall $Q_2=K_{2,t_1}^{*}{\oplus }_{v_1,v_2}{K}_4{\oplus }_{v_1,v_2}T{P}_{2,t_2,(g_1,g_2,\dots ,g_{t_2})}$ with $g_1\ge g_2\ge \dots \ge g_{t_2}\ge 0,M_2=Q_2-v_1{v}_2-{\bigcup }_{i=1}^{t_2}{e}_i$. For an arbitrary $B_1\in \{G\in [M_2,Q_2]:|V\left(G\right)| \ge 6\}$, there is at most one 2-vertex in $B_1$ by Lemma 6. We discuss it in the following four subcases.

Subcase 3.1 $\delta \left(B_1\right)=2$ and $t_1\le 1$.

The possible graphs are either $L_1$ or $L_1\cup \left\{v_1{v}_2\right\}$ shown in the Figure 8. By Lemma 6, $a\left(B_1\right)$ is the 2-vertex. We choose an edge $e=v_3{v}_4$.

Figure 8. Possible graphs of Case 3.

Subcase 3.2 $\delta \left(B_1\right)=2$ and $t_1\ge 2$.

The possible graphs are either $L_2,L_3$ or $L_2\cup \left\{v_1{v}_2\right\},L_3\cup \left\{v_1{v}_2\right\}$ shown in the Figure 8. By Lemma 6, $a\left(B_1\right)$ is the 2-vertex. We choose an edge $e=u_1{u}_2$.

Subcase 3.3 $\delta \left(B_1\right)\ge 3$ and $t_1\ge 4$.

The possible graphs are either $L_4$ or $L_4\cup \left\{v_1{v}_2\right\}$ shown in the Figure 8. We choose an edge either $e=u_1{u}_2$ or $e=u_3{u}_4$ which is not associated with $a\left(B_1\right)$.

Subcase 3.4 $\delta \left(B_1\right)\ge 3$ and $t_1\le 3$.

The possible graphs are either $L_5$ or $L_5\cup \left\{v_1{v}_2\right\}$ shown in the Figure 8. We choose $e=u_1{u}_2$ if it is an interior edge; otherwise, we choose $e=v_3{v}_4$.

For each subcase, we pick the appropriate edge $e=xy$ as discussed above. Let $G^{\prime }=G-e$. By the minimized counterexample property, ${\chi }_{ps}^{\prime }\left(G^{\prime }\left[A_0\right]\right)\le 6\mathsf{\Delta }\left(G^{\prime }\left[A_0\right]\right)$ for any $A_0\subseteq E\left(G^{\prime }\right)$. By Lemma 3 and simple computation, we know $\left|N\right(e\left)\right|\le 2\mathsf{\Delta }\left(G\right[A\left]\right)$ and $d_{G^{\prime }}(x,y)\le 2$. Thus, by Lemma 1, ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 6\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$, a contradiction.

Case 4. $B_1\in \{G\in [M_3,Q_3]:|V\left(G\right)|\ge 6\}$.

Recall that $Q_3=K_{t_1,2}{\oplus }_{v_1,v_2}{K}_5,M_3=Q_3-\{v_1{v}_2,v_1{v}_3,v_2{v}_5\}$. For an arbitrary $B_1\in \{G\in [M_3,Q_3]:|V\left(G\right)|\ge 6\}$. By Lemma 6, we obtain $t_1=1$ and $a\left(B_1\right)=u_1.$ Applying Lemma 4, we have $d_G\left(v\right)=d_{C\left(G\right)}\left(v\right)$ for arbitrary $v\in V\left(B_1\right)\setminus \left\{a\left(B_1\right)\right\}.$ Choose an interior edge $e=v_4{v}_5\in E\left(B_1\right)$ and let $G^{\prime }=G-e.$ Thus we gain $d_{G^{\prime }}(v_4,v_5)\le 2$. By Lemma 5, ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 6\mathsf{\Delta }\left(G\left[A\right]\right)$ is obtained for any $A\subseteq E\left(G\right)$, a contradiction.

Case 5. $B_1$ is a 2-connected $K_4\left(6\right)$-minor free graph.

By Proposition 3, if $B_1\in [K_4\left(5\right),K_5]$, we obtain ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 6\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$ by Case 1 in Section 3.3. If $B_1\in {\cup }_{t_1\ge 1}\{K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4-v_1{v}_2,K_{2,t_1}{\oplus }_{v_1,v_2}{K}_4\}$, by Case 2 in Section 3.3, we have ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 6\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\in E\left(G\right)$. If $B_1$ is a 2-connected $K_4\left(5\right)$-minor free graph, by Case 3 in Section 3.3, we have ${\chi }_{ps}^{\prime }\left(G\left[A\right]\right)\le 6\mathsf{\Delta }\left(G\left[A\right]\right)$ for any $A\subseteq E\left(G\right)$, and the proof is completed.

Based on the proof in Section 3.2–Section 3.4, we prove Theorem 7 when $t=5,6,7$, respectively and complete the proof of Theorem 7.

4. Conclusions

In this paper, we have proved in Theorem 7 that for every $K_4\left(t\right)$-minor free graph with maximum degree $\mathsf{\Delta }\left(G\right)$, ${\chi }_s^{\prime }\left(G\right)\le (t-1)\mathsf{\Delta }\left(G\right)$ for $t\in \{5,6,7\}$. Moreover, these bounds are sharp. This result indicates that the bounds obtained by Batenburg et al. and Wang et al. in [22,23] can be extended to some larger families of graphs. A natural open question is whether similar upper bounds of strong chromaic index exists for $K_4\left(t\right)$-minor free graphs whenever $t>7$. Note that when t goes to infinity, the families of all $K_4\left(t\right)$-minor free graphs would include all graphs.