Edge DP-Coloring in K₄-Minor Free Graphs and Planar Graphs

Abstract

The edge DP-chromatic number of G, denoted by ${\chi }_{DP}^{\prime }\left(G\right)$, is the minimum k such that G is edge DP-k-colorable. In 1999, Juvan, Mohar, and Thomas proved that the edge list chromatic number of $K_4$-minor free graph G with $\Delta \ge 3$ is $\Delta$. In this paper, we prove that if G is a $K_4$-minor free graph, then ${\chi }_{DP}^{\prime }\left(G\right)\in \{\Delta ,\Delta +1\}$, and equality ${\chi }_{DP}^{\prime }\left(G\right)=\Delta +1$ holds for some $K_4$-minor free graph G with $\Delta =3$. Moreover, if G is a planar graph with $\Delta \ge 9$ and with no intersecting triangles, then ${\chi }_{DP}^{\prime }\left(G\right)=\Delta$.

1. Introduction

Graphs considered in this paper are finite and simple. For a graph G and $e,e^{\prime }\in E\left(G\right)$, let $d(e,e^{\prime })$ be the length of the shortest path between the endpoints of e and $e^{\prime }$. The edge chromatic number of G, denoted by ${\chi }^{\prime }\left(G\right)$, is the minimum number of colors needed to color the edges of G so that edges e and $e^{\prime }$ with $d(e,e^{\prime })=0$ are colored by distinct colors. An edge list assignment L assigns to each edge e a set $L\left(e\right)$ of permissible colors. We say G is edge L-colorable if there exists a function $\varphi :E\left(G\right)\to \bigcup L\left(e\right)$ such that

• $\varphi \left(e\right)\in L\left(e\right)$, $\forall e\in E\left(G\right)$;
• $\varphi \left(e\right)\ne \varphi \left(e^{\prime }\right)$, if $d(e,e^{\prime })=0$.

The edge list chromatic number ${\chi }_l^{\prime }\left(G\right)$ is the smallest k such that G is edge L-colorable for every edge list assignment L with $\left|L\left(e\right)\right|\ge k$.

DP-coloring is a generalization of list coloring introduced by Dvořák and Postle [1]. Then, Bernshteyn and Kostochka introduced edge DP-coloring of G as DP-coloring of the line graph of G [2]. To be precise, edge DP-coloring of a graph G is defined as follows:

Definition 1.

Assume G is a graph and $g\in {\mathbb{N}}^{E\left(G\right)}$is a mapping that assigns to each edge e of G a positive integer $g\left(e\right)$. A cover of the line graph of G is a pair $(L,M)$, where $L=\left\{L\right(e):e\in E(G\left)\right\}$ is a family of pairwise disjoint sets, and $M=\{M_{e{e}^{\prime }}:e\ne e^{\prime },d(e,e^{\prime })=0\}$ is a family of bipartite graphs such that $M_{e{e}^{\prime }}$ has the bipartite set $L\left(e\right)$ and $L\left(e^{\prime }\right)$ and $\Delta \left(M_{e{e}^{\prime }}\right)\le 1$. An g-cover is a cover $(L,M)$ such that $\left|L\right(e\left)\right|\ge g\left(e\right)$ for each edge e. An $(L,M)$-edge coloring of G is a mapping ϕ such that $\varphi \left(e\right)\in L\left(e\right)$ for each edge e, and for any pair of edges $e,e^{\prime }$ with $d(e,e^{\prime })=0$, $\{\varphi \left(e\right),\varphi \left(e^{\prime }\right)\}\notin M_{e{e}^{\prime }}$ (for convenience, we write $M_{e{e}^{\prime }}$ for $E\left(M_{e{e}^{\prime }}\right)$).

Definition 2.

If G has an $(L,M)$-edge coloring for every g-cover $(L,M)$, then we say G is edge DP-g-colorable. If $g\left(e\right)=k$ for each edge e, then edge DP-g-colorable is called edge DP-k-colorable. The edge DP-chromatic number of G, denoted by ${\chi }_{DP}^{\prime }\left(G\right)$, is the minimum integer k such that G is edge DP-k-colorable.

It is easy to see and well known that for any graph G, ${\chi }^{\prime }\left(G\right)\le {\chi }_l^{\prime }\left(G\right)\le {\chi }_{DP}^{\prime }\left(G\right)$.

A graph G is $K_4$-minor free if it has no subgraph isomorphic to a subdivision of $K_4$. There are many good results based on the $K_4$-minor free graphs. Beaudou, Foucaud, and Naserasr [3] studied the homomorphism of $K_4$-minor free graphs with odd girth. Combined with results from [4,5], it is obvious that every $K_4$-minor free graph is 2-degenerate, and its strong edge chromatic number is at most $6\Delta -7$. Meanwhile, Juvan, Mohar, and Thomas proved the following theorem.

Theorem 1

([6]). Let G be a $K_4$-minor free graph with $\Delta \ge 3$. Then ${\chi }_l^{\prime }\left(G\right)=\Delta$.

The following conjecture is known as the edge list coloring conjecture, which was proposed by several researchers (see [7,8]).

Conjecture 1.

${\chi }_l^{\prime }\left(G\right)={\chi }^{\prime }\left(G\right)$ for a loopless multigraph G.

The conjecture is verified for bipartite multigraphs [9], complete graphs of odd order [10], and complete graphs $K_{p+1}$ where p is an odd prime [11], and remains largely open in general.

Vizing’s theorem implies that for every simple graph, the edge chromatic number is at most $\Delta +1$. The next conjecture is a combination of Conjecture 1 and Vizing’s theorem.

Conjecture 2.

${\chi }_l^{\prime }\left(G\right)\le \Delta +1$ for a simple graph G.

Borodin [12] proved Conjecture 2 for planar graphs G when $\Delta \left(G\right)\ge 9$, and Bonamy [13] improved this result to planar graphs with $\Delta \ge 8$.

Theorem 2

([12]). If G is a planar graph with $\Delta \ge 9$, then ${\chi }_l^{\prime }\left(G\right)\le \Delta +1.$

Theorem 3

([13]). If G is a planar graph with $\Delta \ge 8$, then ${\chi }_l^{\prime }\left(G\right)\le \Delta +1.$

By Theorem 3, we can see that if G is a planar graph with $\Delta \ge 8$, then ${\chi }_l^{\prime }\left(G\right)\in \{\Delta ,\Delta +1\}.$ Moreover, edge DP-coloring and DP-coloring of graphs are also studied in the literature [14,15,16]. In particular, the following two results were proved by Zhang et al. in [16].

Theorem 4.

Assume that G is a planar graph with maximum degree Δ such that G has no cycle of length k for $k\in \{3,4\}$. Then, ${\chi }_{DP}^{\prime }\left(G\right)=\Delta$ if either $\Delta \ge 7$ and $k=4$, or $\Delta \ge 8$ and $k=3$.

Theorem 5.

If G is a planar graph with maximum degree $\Delta \ge 9$, then ${\chi }_{DP}^{\prime }\left(G\right)\le \Delta +1$.

In this paper, we first study edge DP-coloring of $K_4$-minor free graphs by following the theorem and constructing a $K_4$-minor free graph with $\Delta =3$, which is not edge DP-3-colorable.

Theorem 6.

Let G be a $K_4$-minor free graph with maximum degree Δ. Then, ${\chi }_{DP}^{\prime }\left(G\right)\le \Delta +1$.

Then, consideration of the edge DP-chromatic number of planar graph G with $\Delta \ge 9$ as an improvement of Theorem 5 is given by following Corollary 1, which is implied from Theorem 7.

Theorem 7.

If G is a planar graph with maximum degree Δ, and with no intersecting triangles, then ${\chi }_{DP}^{\prime }\left(G\right)=max\{\Delta ,9\}$.

Corollary 1.

If G is a planar graph with $\Delta \ge 9$ and with no intersecting triangles, then ${\chi }_{DP}^{\prime }\left(G\right)=\Delta$.

Compared with Theorems 5 and 7, the discharging method, as a powerful tool for graph-coloring problems, is also used to consider reducible configurations. By our result, we give one sufficient condition for planar graph G with $\Delta \ge 9$ such that ${\chi }_{DP}^{\prime }\left(G\right)=\Delta$.

Our paper is organized as follows. In Section 2, some lemmas that are used in the proof of Theorems 6 and 7 are listed. In Section 3, we complete the proof of Theorem 6, and a $K_4$-minor free graph with $\Delta =3$ that is not edge DP-3-colorable is given, corresponding the matching assignment ${\mathcal{M}}_L$, which is described in this part. In Section 4, the proof of Theorem 7 is given. Finally, in the Section 5, the conclusions of this paper are stressed.

2. Some Preliminaries

In this section, we introduce some lemmas that are used in the proof of our results. It is well known [6] that a $K_4$-minor free graph has some special structure, as shown in the lemma below.

Lemma 1

([6]). There exists one of the following structures in every $K_4$-minor free graph G:

(a) 

A vertex of degree at most one;

(b) 

Two distinct vertices of degree two with the same neighbors;

(c) 

Two distinct vertices u,v and not necessarily distinct vertices $w, z\in V\left(G\right)\setminus \{u, v\}$ such that the neighbors of v are u and w, and every neighbor of u is equal v, w, or z;

(d) 

Five distinct vertices $v_1$,$v_2$,$u_1$,$u_2$,w such that the neighbors of w are $u_1$,$u_2$,$v_1$,$v_2$, and the neighbors of $v_i$ are w and $u_i$ for $i=1,2$.

Below, we provide a special edge list assignment L and show that the graph of $\left(d\right)$ (as shown in Figure 1) in Lemma 1 is edge DP-L-colorable.

Figure 1. Graph and the value of its list assignment of (d) in Lemma 1.

Lemma 2.

Let H be the graph in Figure 1, and L be an edge list assignment with $\left|L\left(e\right)\right|\ge 3$ if $e\in \{v_1{u}_1,v_2{u}_2,u_{1}w,u_{2}w\}$, and $\left|L\left(e\right)\right|\ge 5$ if $e\in \{v_{1}w,v_{2}w\}$. Then, H has an $(L,M)$-edge coloring for any cover $(L,M)$.

Proof. 

The proof is trivial since H can be colored greedily by the order of $u_{1}w,u_{2}w,u_1{v}_1,$$v_{1}w,v_{2}w,v_2{u}_2$. □

Let G be a graph with a cover $(L,M)$. Suppose that H is a subgraph of G and $G^{\prime }=G-E\left(H\right)$ has an $(L^{\prime },M^{\prime })$-edge coloring with

$$L^{\prime }=\{L^{\prime }\left(e\right)=L\left(e\right)\in L:e\in E\left(G^{\prime }\right)\}\mathrm{and}{M}^{\prime }=\{M_{e{e}^{\prime }}^{\prime }=M_{e{e}^{\prime }}\in M:e{\sim }_{G^{\prime }}{e}^{\prime }\}.$$

There is a mapping ${\varphi }^{\prime }$ such that ${\varphi }^{\prime }\left(e\right)\in L^{\prime }\left(e\right)$ for each edge e in $G^{\prime }$, and for any pair of edges $e,e^{\prime }$ with $d_{G^{\prime }}(e,e^{\prime })=0$, $\{{\varphi }^{\prime }\left(e\right),{\varphi }^{\prime }\left(e^{\prime }\right)\}\notin M_{e{e}^{\prime }}^{\prime }$.

For $e\in E\left(H\right)$, we define a new list assignment $L^{*}\left(e\right)$ and a new family of bipartite graphs $M^{*}$ as below:

$$L^{*}\left(e\right)=L\left(e\right)\setminus \bigcup _{e^{\prime }\sim e}\{c\in L\left(e\right):\exists {\varphi }^{\prime }\left(e^{\prime }\right)\in L^{\prime }\left(e^{\prime }\right)\mathrm{s}.\mathrm{t}.\{c,{\varphi }^{\prime }\left(e^{\prime }\right)\}\in M_{e{e}^{\prime }}\in M\}$$

and

$$M_{e{e}^{\prime }}^{*}=\{\{c,c^{\prime }\}\in M_{e{e}^{\prime }}\in M:c\in L^{*}\left(e\right),c^{\prime }\in L^{*}\left(e^{\prime }\right)\},\forall e{\sim }_H{e}^{\prime }.$$

If $G^{\prime }$ has an $(L^{\prime },M^{\prime })$-edge coloring ${\varphi }^{\prime }$ and H has an $(L^{*},M^{*})$-edge coloring ${\varphi }^{*}$, then ${\varphi }^{\prime }\cup {\varphi }^{*}$ is an $(L,M)$-edge coloring of G. Hence, G has an $(L,M)$-edge coloring. This gives the following lemma, proved in [16].

Lemma 3

([16]). Let G be a graph with a cover $(L,M)$, and H be a subgraph of G. If $G-E\left(H\right)$ has an $(L^{\prime },M^{\prime })$-edge coloring and H has an $(L^{*},M^{*})$-edge coloring, then G has an $(L,M)$-edge coloring, where $L^{\prime }$, $M^{\prime }$, $L^{*}$, and $M^{*}$ are defined as above.

Besides, Zhang et al. [16] provided the following lemmas as powerful tools to study the edge DP-coloring of planar graphs. Let G be a graph; a vertex v of G is a pendant vertex if v has degree 1. Similarly, an edge of a graph is said to be pendant if one of its endpoints is a pendant vertex.

Lemma 4

([16]). Let G be a cycle with a pendant edge, and L be an edge list assignment of G satisfying $\left|L\left(uv\right)\right|\ge d\left(u\right)+d\left(v\right)-2$ for every $uv\in E\left(G\right)$. Then, G has an $(L,M)$-edge coloring for any cover $(L,M)$, where M is a family of bipartite graphs over L.

Lemma 5

([16]).　Let $G=C+\{v_1{v}_{2i}:i\in [2,t-1]\}+v_{1}u$($t\ge 3$), where $C=v_1{v}_2\dots v_{2t}{v}_1$ is a cycle and $v_{1}u$ is a pendant edge. If L is an edge list assignment of G satisfying $\left|L\left(v_{1}u\right)\right|\ge t$, $\left|L\left(v_1{v}_{2i}\right)\right|\ge t+1$ for $i\in [1,t]$, $\left|L\right(e\left)\right|\ge 2$ for other edges e of G, then G has an $(L,M)$-edge coloring for any cover $(L,M)$, where M is a family of bipartite graphs over L.

Before the next section, some notations need to be introduced in advance. Let G be a graph and f be a k-face; all vertices of f will be ordered as $v_1,v_2\dots ,v_k$ clockwise, and denoted by $[v_1{v}_2\dots v_k]$. The definitions of all symbols used are shown clearly by the following table.

The Definitions of All Symbols
Symbol	Definition
$V\left(G\right)$	The set of all vertices in G
$F\left(G\right)$	The set of all faces in G
k($k^{+}$; $k^{-}$)-vertex	A vertex of degree k (at least k; at most k)
k($k^{+}$; $k^{-}$)-face	A face of length k (at least k; at most k)
$(d_1,d_2,\dots ,d_k)$-face	All vertices of face will be ordered as $v_1,v_2\dots ,v_k$ clockwise and for each $i\in \left[k\right]$, $v_i$ is $d_i$-vertex
$(d_1,d_2)$-edge	$v_1{v}_2$ is a $(d_1,d_2)$-edge if $v_i$ has degree $d_i$ for $i\in \left[2\right]$

3. Edge DP-Coloring of $K_4$-Minor Free Graph

In this section, we provide a proof of Theorem 6 and construct a $K_4$-minor free graph with $\Delta =3$ that is not edge DP-3-colorable.

Theorem 8.

Let G be a $K_4$-minor free graph with maximum degree Δ; then, ${\chi }_{DP}^{\prime }\left(G\right)\le \Delta +1$.

Proof. 

Assume that G is a counterexample of Theorem 6 with $\left|V\right(G\left)\right|+\left|E\right(G\left)\right|$ minimal. Let $(L,M)$ be a cover with $\left|L\right(e\left)\right|=\Delta +1$ for all $e\in E\left(G\right)$ such that G does not have an $(L,M)$-edge coloring. As G is a $K_4$-minor free graph, G contains one of the structures (a–d) in Lemma 1.

(a) Assume that $u\in V\left(G\right)$ with $e=uv\in E\left(G\right)$ and $d\left(u\right)=1$. Then, $G^{\prime }=G-e$ has an $(L^{\prime },M^{\prime })$-edge coloring by minimality, where $L^{\prime }$ and $M^{\prime }$ are defined in Lemma 3. Note that e has at most $\Delta -1$ incident edges in $G^{\prime }$. Thus, we have $|L^{*}\left(e\right)|\ge 2$, and so e can be colored properly. Then, G has an $(L,M)$-edge coloring by Lemma 3, a contradiction.

(b) Let $u,v\in V\left(G\right)$ such that $N\left(u\right)=N\left(v\right)=\{w,y\}$ (it may happen that $uv\in E\left(G\right)$ and $N\left(u\right)=\{v,w\},N\left(v\right)=\{u,w\}$). Set $G^{\prime }=G-E\left(C\right)$, where $C=uwvyu$ (or $C=uvw$). By minimality, $G^{\prime }$ has an $(L^{\prime },M^{\prime })$-edge coloring. And C has an $(L^{*},M^{*})$-edge coloring because of $|L^{*}\left(e\right)|\ge 3$, $\forall e\in E\left(C\right)$. Similarly, G has an $(L,M)$-edge coloring by Lemma 3, a contradiction.

(c) If $w=z$, then this situation is included in (a) or (b). Assume that $w\ne z$. Then, $G-uv$ has an $(L^{\prime },M^{\prime })$-edge coloring by minimality. Note that $d\left(v\right)=2$ and $v\in N\left(u\right)\subseteq \{v,w,z\}$. Thus,

$$\begin{array}{cc}\hfill |L^{*}\left(uv\right)|& =\Delta +1-\left(d\right(u)-1)-\left(d\right(v)-1)\ge max\left\{d\right(u),d(v\left)\right\}-d\left(u\right)-d\left(v\right)+3\hfill \\ \hfill & \ge max\left\{d\right(u),d(v\left)\right\}-d\left(u\right)+1\ge 1.\hfill \end{array}$$

So $uv$ could be colored directly. By Lemma 3, G could have an $(L,M)$-edge coloring, a contradiction.

(d) Let H be the graph in Figure 1. Then $G^{\prime }=G-E\left(H\right)$ has an $(L^{\prime },M^{\prime })$-edge coloring by minimality. It is easy to check that for all $e\in E\left(H\right)$, $|L^{*}\left(e\right)|$ satisfies the condition in Lemma 2. Thus, H has an $(L^{*},M^{*})$-edge coloring, and so G has an $(L,M)$-edge coloring, a contradiction. □

In the following, we will introduce a $K_4$-minor free graph G with a maximum degree of 3, which is not edge DP-3-colorable. In particular, we will define a $K_4$-minor free graph G, a cover $(L,M)$ with $\left|L\right(e\left)\right|=3$, $\forall e\in E\left(G\right)$ such that G has no $(L,M)$-edge coloring.

Definition 3.

Let $(L,M)$ be a cover of G and $(e_i,e_j)$ be an adjacent pair. We call the cover M straight over $(e_i,e_j)$ if every $\left\{(e_i,c_1)(e_j,c_2)\right\}\in M_{e_i{e}_j}$ satisfies $c_1=c_2$. Especially, for edge set E, we call M straight over E if M is straight over every adjacent pair $(e_i,e_j)$, $\forall e_i,e_j\in E$.

Lemma 6.

Let H be the graph in Figure 2a with an edge list assignment L, where $L\left(e\right)=\{b,c\}$ for $e\in \{uw,uz\}$, and $L\left(e\right)=\{a,b,c\}$ otherwise. Then, there exists a family of bipartite graphs $M^{\prime }$ over L such that H does not have any $(L,M^{\prime })$-edge coloring.

Figure 2. Graph H and a cover $(M^{\prime },L)$ of H.

Proof. 

Let $M^{\prime }$ be the family of bipartite graphs over L, as shown in Figure 2b, which is straight over $(uw,uz),(wv,wz),(wv,vz)$ and $(vz,wz)$. Assume that H has an $(L,M^{\prime })$-edge coloring $\varphi$. Without loss of generality, we may assume that $\varphi \left(uw\right)=b$. Then, the only choice for edge $uz,wz,vz$ is $c,c,b$, respectively. Now we cannot find available colors for $wv$. Similarly, we can get a contradiction if $\varphi \left(uw\right)=c$. Therefore, H does not have an $(L,M^{\prime })$-edge coloring. □

Lemma 7.

Let G be a $K_4$-minor free graph as shown in Figure 3. Then, G is not edge DP-3-colorable.

Figure 3. K₄-minor free graph G.

Proof. 

Let $L\left(e\right)=\{a,b,c\}$ for each $e\in E\left(G\right)$. Set $E_0={\displaystyle \bigcup _{i=1,2,3}}\{x{u}_i,u_i{w}_i,u_i{z}_i\}$, $E_i=\{u_i{w}_i,u_i{z}_i,w_i{z}_i,w_i{v}_i,v_i{z}_i\}$ and $H_i=G\left[E_i\right]$ for $i\in \left[3\right]$. Note that $H_i$ is a copy of H in Lemma 6. Now, we will define a family of bipartite graphs M over L such that G has no $(L,M)$-edge coloring. Let M be a family of bipartite graphs over L such that

• M is straight over $E_0$;
• For $i\in \left[3\right]$, $M_{e{e}^{\prime }}=M_{e{e}^{\prime }}^{\prime }$, $\forall e,e^{\prime }\in E_i=\{u_i{w}_i,u_i{z}_i,w_i{z}_i,w_i{v}_i,v_i{z}_i\}$ ($M^{\prime }$ is shown in Figure 2b).

Assume that G has an $(L,M)$-edge coloring $\varphi$. As M is straight over $\{x{u}_1,x{u}_2,x{u}_3\}$, there must exist exactly one edge $x{u}_j,j\in \{1,2,3\}$, say $j=1$ s.t. $\varphi \left(x{u}_1\right)=a$. Then a is not available for $u_1{w}_1$ and $u_1{z}_1$. Thus, for graph $H_1$, the remaining list assignment $L^{\prime }$ satisfies $L^{\prime }\left(e\right)=\{b,c\}$ for $e\in \{u_1{w}_1,u_1{z}_1\}$ and $L^{\prime }\left(e\right)=\{a,b,c\}$ otherwise. By Lemma 6, $H_1$ does not have $(L^{\prime },M^{\prime })$-edge coloring, and so G does not have an $(L,M)$-edge coloring. Therefore, G is not edge DP-3-colorable. □

4. Proof of Theorem 7

Assume G is a counterexample to Theorem 7 such that $\left|E\left(G\right)\right|$ is minimal. Then, there exists a cover $(L,M)$ with $\left|L\left(e\right)\right|=\Delta$ for $e\in E\left(G\right)$ such that G has no $(L,M)$-edge coloring.

The lemma below shows some properties of the minimal counterexample G. We say f is a special 4-face with facial cycle $\left[v_1{v}_2{v}_3{v}_4\right]$ if either (Type I) $d\left(v_1\right)=d\left(v_3\right)=2$, $d\left(v_2\right)=d\left(v_4\right)=\Delta$ or (Type II) $d\left(v_1\right)=2,d\left(v_3\right)=3$, $d\left(v_2\right)=d\left(v_4\right)=\Delta$. Let ${\mathcal{F}}_1$ (${\mathcal{F}}_2$) be the set of special faces of Type I (Type II), respectively. Figure 4 describes these two kinds of special faces.

Figure 4. Two kinds of special 4-faces.

Lemma 8.

If G is a counterexample to Theorem 7 with $\left|E\left(G\right)\right|$ minimal, then all of following hold:

(a) 

G is connected.

(b) 

Each $v\in V\left(G\right)$ is incident to at most one triangle.

(c) 

$d\left(u\right)+d\left(v\right)\ge \Delta +2$ for any edge $uv\in E\left(G\right)$.

(d) 

If $d\left(v\right)=\Delta$ and v is incident to some $f\in {\mathcal{F}}_1$, then all vertices in $N_G\left(v\right)\setminus V\left(f\right)$ are $3^{+}$-vertex.

(e) 

G does not contain $H_0$ (shown in Figure 5a) as its subgraph, where Δ-vertex $v_1$ is incident to a 2-vertex u and two special faces of Type II.

Figure 5. Two configurations of (e) and (f) in Lemma 8.

(f) 

G does not contain $H_1$(shown in Figure 5b) as its subgraph, where Δ-vertex v is incident to a 2-vertex $u_0$ and three 4-faces. Moreover, $\left[u_1{u}_{3}v{u}_2\right]$ and $\left[w_{2}v{w}_3{w}_1\right]$ are two special faces of Type II and $\left[v{u}_{3}x{w}_3\right]$ shares a common $(\Delta ,3)$-edge with two special faces. Here, $u_1$ and $w_1$ are not necessarily different.

Proof. 

(a,b) It is trivial, since G is a minimal counterexample and does not have intersecting triangles.

(c) Assume that there is an edge $e=uv$ of G satisfying $d\left(u\right)+d\left(v\right)\le \Delta +1$. Let $G^{\prime }=G-\left\{uv\right\}$. By minimality, $G^{\prime }$ has an $(L^{\prime },M^{\prime })$-edge coloring for cover $(L^{\prime },M^{\prime })$, the definition of $L^{\prime }$ and $M^{\prime }$ are given in Section 2. Note that $\left|L^{*}\left(e\right)\right|\ge \Delta -(d\left(u\right)+d\left(v\right)-2)\ge 1$. Thus, e could be colored. By Lemma 3, G has an $(L,M)$-edge coloring, a contradiction.

(d) Let $v\in V\left(G\right)$ with $d\left(v\right)=\Delta$ and $f\in {\mathcal{F}}_1$ with $f=\left[v_{1}v{v}_3{v}_4\right]$. Note that $d\left(v_1\right)=d\left(v_3\right)=2$ and $d\left(v_4\right)=\Delta$. Assume there exists $u\in N_G\left(v\right)\setminus V\left(f\right)$ with $d\left(u\right)\le 2$. Let H be a subgraph of G with $V\left(H\right)=V\left(f\right)+\left\{u\right\}$ and $E\left(H\right)=E\left(f\right)+\left\{vu\right\}$. Then, by the minimality of G, $G-E\left(H\right)$ has an $(L^{\prime },M^{\prime })$-edge coloring ${\varphi }^{\prime }$. Now we consider $|L^{*}\left(e\right)|$ for $e\in E\left(H\right)$. It is easy to check that $|L^{*}\left(vu\right)|\ge \Delta -(1+\Delta -3)\ge 2$, $|L^{*}\left(e\right)|\ge \Delta -(\Delta -2)=2$ for $e\in \{v_3{v}_4,v_4{v}_1\}$, $|L^{*}\left(e\right)|\ge \Delta -(\Delta -3)=3$ for $e\in \{v_{1}v,v{v}_3\}$. By Lemma 4, H has an $(L^{*},M^{*})$-edge coloring ${\varphi }^{*}$. Since $G-E\left(H\right)$ has an $(L^{\prime },M^{\prime })$-edge coloring ${\varphi }^{\prime }$, G has an $(L,M)$-edge coloring ${\varphi }^{\prime }\cup {\varphi }^{*}$.

(e) Assume $H_0\subseteq G$. Then, $G-E\left(H_0\right)$ has an $(L^{\prime },M^{\prime })$-edge coloring. Note that $H_0$ is a structure described in Lemma 5 with $t=3$. Based on its degree condition, it is easy to check that $\left|L^{*}\left(v_{1}u\right)\right|\ge \Delta -(\Delta -4+1)=3$, $\left|L^{*}\left(v_1{v}_2\right)\right|=\left|L^{*}\left(v_1{v}_6\right)\right|=\left|L^{*}\left(v_1{v}_4\right)\right|\ge \Delta -(\Delta -4)=4$, and $\left|L^{*}\left(e\right)\right|\ge \Delta -(\Delta -2)=2$ for $e\in E\left(H_0\right)$ otherwise. By Lemma 5, $H_0$ has an $(L^{*},M^{*})$-edge coloring. Thus, G has an $(L,M)$-edge coloring by Lemma 3, which is a contradiction.

(f) Observe that $H_1$ is the structure of Lemma 5 with $t=4$ and $d\left(x\right)\le \Delta$. Thus, we have $|L^{*}\left(e\right)|\ge 2$ for $e\in \{u_1{u}_2,u_1{u}_3,w_1{w}_2,w_1{w}_3,x{u}_3,x{w}_3\}$, $|L^{*}\left(e\right)|\ge 5$ for $e\in \{v{u}_2,v{w}_2,v{u}_3,v{w}_3\}$ and $|L^{*}\left(v{u}_0\right)|\ge 4$. Thus, $H_1$ has an $(L^{*},M^{*})$-edge coloring, and so G has an $(L,M)$-edge coloring by Lemma 3, which is a contradiction. □

To drive a contradiction by discharging analysis, we first define an initial charge $ch$ as $ch\left(v\right)=2d\left(v\right)-6$ for $v\in V\left(G\right)$ and $ch\left(f\right)=d\left(f\right)-6$ for $f\in F\left(G\right)$.

By Euler’s formula $\left|V\right(G\left)\right|-\left|E\right(G\left)\right|+\left|F\right(G\left)\right|=2$, the total sum of charges of vertices and faces satisfies the following identity:

$$\sum _{x\in V\left(G\right)\cup F\left(G\right)}ch\left(x\right)=\sum _{v\in V\left(G\right)}(2d\left(v\right)-6)+\sum _{x\in F\left(G\right)}(d\left(f\right)-6)=-12.$$

Next, we design appropriate discharging rules and redistribute charges accordingly. Once the discharging is finished, a new charge $c{h}^{*}$ is produced. Note that the discharging process preserves the total sum of charges of G. However, we will show that $c{h}^{*}\left(x\right)\ge 0$ for all $x\in V\left(G\right)\cup F\left(G\right)$, which leads to an obvious contradiction, and subsequently the proof is complete.

Our discharging rules are defined as follows. (For $3,4,5$-face f, we always assume $v_1\in V\left(f\right)$ has the minimal degree).

(R1)

Every 2-vertex v receives 1 from every incident face.

(R2)

If a $\Delta$-vertex v is incident to a $6^{+}$-face f; and

(R2.1)

v is adjacent to two 2-vertices which are incident to f, then v gives 1 to f.

(R2.2)

v is adjacent to exactly one 2-vertex which is incident to f, then v gives $\frac{1}{2}$ to f.

(R3)

Let $f=\left[v_1{v}_2{v}_3\right]$ be a 3-face.

(R3.1)

If $d\left(v_1\right)=2$, then each of $v_2$ and $v_3$ gives 2 to f, respectively.

(R3.2)

If $d\left(v_1\right)=3$, then each of $v_2$ and $v_3$ gives $\frac{3}{2}$ to f, respectively.

(R3.3)

If $d\left(v_1\right)=4$, then each of $v_2$ and $v_3$ gives $\frac{5}{4}$ to f; $v_1$ gives $\frac{1}{2}$ to f.

(R3.4)

If $d\left(v_1\right)=5$,

• and $\exists i\in \{2,3\}$ with $d\left(v_i\right)=\Delta -3$, then each of $v_1,v_2$ and $v_3$ gives 1 to f.
• otherwise, $v_1$ gives $\frac{1}{2}$ to f, each of $v_2$ and $v_3$ gives $\frac{5}{4}$ to f.

(R3.5)

If $d\left(v_1\right)\ge 6$, then each of $v_1,v_2$ and $v_3$ gives 1 to f.

(R4)

Let $f=\left[v_1{v}_2{v}_3{v}_4\right]$ be a 4-face.

(R4.1)

If $d\left(v_1\right)=2$ (note that $d\left(v_2\right)=d\left(v_4\right)=\Delta$), and

• $d\left(v_3\right)=2$, then each of $v_2$ and $v_4$ gives 2 to f ($f\in {\mathcal{F}}_1$ due to (c) in Lemma 8).
• $d\left(v_3\right)=3$, then each of $v_2$ and $v_4$ gives $\frac{3}{2}$ to f ($f\in {\mathcal{F}}_2$).
• $d\left(v_3\right)=4$, then each of $v_2$ and $v_4$ gives $\frac{5}{4}$ to f and $v_3$ gives $\frac{1}{2}$ to f.
• $d\left(v_3\right)=5$, then each of $v_2$ and $v_4$ gives $\frac{11}{10}$ to f and $v_3$ gives $\frac{4}{5}$ to f.
• $d\left(v_3\right)\ge 6$, then each of $v_2$ and $v_4$ gives 1 to f, $v_3$ gives 1 to f.

(R4.2)

If $d\left(v_1\right)=3$, and

• $d\left(v_3\right)=3$, then each of $v_2$ and $v_4$ gives 1 to f.
• $d\left(v_3\right)\ge 4$, then each of $v_2$ and $v_4$ gives $\frac{3}{4}$ to f and $v_3$ gives $\frac{1}{2}$ to f.

(R4.3)

If $d\left(v_1\right)\ge 4$, then each of $v_1,v_2,v_3$ and $v_4$ gives $\frac{1}{2}$ to f.

(R5)

Let $f=\left[v_1{v}_2{v}_3{v}_4{v}_5\right]$ be a 5-face. By (c) in Lemma 8 and symmetry, we could assume $v_3$ has the second smallest degree.

(R5.1)

If $d\left(v_1\right)=2$,

• and $d\left(v_3\right)=2$, then each of $v_2,v_4$ and $v_5$ gives 1 to f.
• and $d\left(v_3\right)=3$, then each of $v_2$ and $v_4$ gives $\frac{1}{2}$ to f, $v_5$ gives 1 to f.
• otherwise, each of $v_2,v_3,v_4$ and $v_5$ gives $\frac{1}{2}$ to f.

(R5.2)

If $d\left(v_1\right)=3$,

• and $d\left(v_3\right)=3$, then each of $v_2,v_4$ and $v_5$ gives $\frac{1}{3}$ to f.
• otherwise, each of $v_2,v_3,v_4$ and $v_5$ gives $\frac{1}{4}$ to f.

(R5.3)

If $v_1$ is a $4^{+}$-vertex, then each $v_i$ gives $\frac{1}{5}$ to f for $i\in \left[5\right]$.

Note that each face f only gives a charge to incident 2-vertex. Assume that $\left|f\right|\ge 6$. If there is a 2-vertex $v\in V\left(f\right)$, then there exist two $\Delta$-vertices as its neighbors on f by (c) in Lemma 8. Thus, by (R1) and (R2), each $\Delta$-vertices gives at least $1/2$ to f, and so $c{h}^{*}\left(f\right)\ge ch\left(f\right)\ge 0$. If f is a d-face, $d\in \{3,4,5\}$, then $c{h}^{*}\left(f\right)=0$ by rules (R3), (R4) and (R5). After all, the final charge of every face is nonnegative.

As $d\left(u\right)+d\left(v\right)\ge \Delta +2$ for any edge $uv\in E\left(G\right)$ (by (c) of Lemma 8), each v of G has degree at least 2. Now, we will consider the final charge of d-vertex, where $d\in [2,\Delta ]$. If $d\left(v\right)=2$, then $ch\left(v\right)=-2$. As v does not give out any charge, $c{h}^{*}\left(v\right)=-2+1\times 2=0$ by rule (R1). Now it suffices to consider the vertex v with $d\left(v\right)\in [3,\Delta ]$.

Claim 1.

If v is d-vertex with $d\in [3,\Delta -1]$, then $c{h}^{*}\left(v\right)\ge 0$.

Proof. 

If $d\left(v\right)=3$, then v does not give out or get any charge by all discharging rules. So, $c{h}^{*}\left(v\right)=ch\left(v\right)=0$.

If $d\left(v\right)=4$, then v gives at most $\frac{1}{2}$ to every incident face by rules (R3), (R4) and (R5). So, $c{h}^{*}\left(v\right)\ge 2-4\times \frac{1}{2}=0$.

Next considering any 5-vertex v. If v is not incident to any 3-face, then by rules (R4) and (R5), we obtain $c{h}^{*}\left(v\right)\ge 4-5\times \frac{4}{5}=0$. Otherwise, v is incident to exactly one 3-face by (b) in Lemma 8. For $i\in \left[5\right]$, let $u_i\in N\left(v\right)$ in cyclic order and for $i\in \left[5\right]$, let $f_i$ be the face incident to v with $v{u}_i,v{u}_{i+1}$, where indices are taken module 5. Assume that the unique 3-face is $f_1$. If v gives at most $\frac{4}{5}$ to $f_1$, then $c{h}^{*}\left(v\right)\ge 0$ obviously. So, considering v gives 1 to $f_1$ by rule (R3.4). Then, $\exists i\in \{1,2\}$ such that $d\left(u_i\right)=\Delta -3$, and so, v gives at most $\frac{1}{2}$ to the face $f\in \{f_2,f_5\}$ with $u_i\in V\left(f\right)$ by (R4.3) and (R5). Thus, $c{h}^{*}\left(v\right)\ge 4-1-\frac{1}{2}-3\times \frac{4}{5}\ge 0$.

If $d\left(v\right)=6$, then v sends at most 1 to every incident face by (R3), (R4), and (R5). So, $c{h}^{*}\left(v\right)\ge 6-1\times 6=0$.

If $d\left(v\right)\in [7,\Delta -1]$, by (R3), (R4) and (R5), then v sends at most $\frac{3}{2}$ to 3-face, and v sends at most 1 to every incident $4^{+}$-face. Therefore, we obtain $c{h}^{*}\left(v\right)\ge 2d-6-\frac{3}{2}\times 1-(d-1)\times 1=d-6.5>0$. □

Claim 2.

If v is a $\Delta$-vertex, then $c{h}^{*}\left(v\right)\ge 0$.

Proof. 

For $i\in [\Delta ]$, let $u_i\in N\left(v\right)$ in cyclic order and $f_i$ be the face incident to v with $v{u}_i,v{u}_{i+1}$, where indices are taken module $\Delta$. Let $F=\{f_i:i\in [\Delta ]\}$. We call $P\subseteq F$ is a petal of v if P is formed by some consecutive adjacent $f\in {\mathcal{F}}_2$ and set $l=max\left\{\right|P|:P\mathrm{is}\mathrm{a}\mathrm{petal}\mathrm{of}v\}$.

If $l\ge 5$, then there is some subgraph $H_0$ of G, which contradicts (e) of Lemma 8. Thus, we can have the following observation.

Observation 1.

Let w be the number of petals around v. If ${\mathcal{F}}_2\cap F\ne \varnothing$, i.e., $w\ge 1$, then $l\le 4$ and one of the following situations appears.

(s1) 

$l=4$, $\omega =1$ (as shown in Figure 6a);

Figure 6. (s1) and (s2) in Observation 1.

(s2) 

$l=3$, $\omega =1$ (as shown in Figure 6b);

(s3) 

$l=2$ and two consecutive adjacent special faces share common $(\Delta ,3)$-edge, $\omega =1$;

(s4) 

$l=2$ and two consecutive adjacent special faces share common $(\Delta ,2)$-edge, $\omega \le ⌊\frac{\Delta }{2}⌋$;

(s5) 

$l=1$, $\omega \le ⌊\frac{\Delta }{2}⌋$.

Note that if there is some 3-face in F, there is exactly one 3-face in F since there are no intersecting triangles in G. Now we will consider three cases based on the existence of special faces in F. Assume that ${\mathcal{F}}_i\cap F=\varnothing$ for $i\in \left[2\right]$. By the (R2), (R3), (R4) and (R5), v sends at most 2 to the possible unique 3-face and at most $\frac{5}{4}$ to $4^{+}$-faces. Thus, $c{h}^{*}\left(v\right)\ge 2\Delta -6-2-\frac{5}{4}(\Delta -1)\ge 0$ since $\Delta \ge 9$. Assume that there is some special 4-face of Type I in F, say $f_1\in {\mathcal{F}}_1\cap F$. By (d) in Lemma 8, the degree of $u_i$ is at least 3, for $i\in \{3,4,\dots ,\Delta \}$. If one of $f_2$ and $f_{\Delta }$ is 3-face, say $f_2$, then v sends 2 to $f_1$, at most 2 to $f_2$, at most $\frac{3}{2}$ to $f_{\Delta }$, and at most 1 to $f_i,i\in [3,4,\dots ,\Delta -1]$ by rules (R3.5). So, $c{h}^{*}\left(v\right)\ge 2\Delta -6-2-2-\frac{3}{2}-1\times (\Delta -3)>0$ since $\Delta \ge 9$. Otherwise, v sends 2 to $f_1$, at most $\frac{3}{2}$ to $f_2$, $f_{\Delta }$ and possible 3-face, and at most 1 to other $f_i\in F$. So, $c{h}^{*}\left(v\right)\ge 2\Delta -6-2-3\times \frac{3}{2}-1\times (\Delta -4)\ge 0$.

In the following, we will assume that $F\cap {\mathcal{F}}_1=\varnothing$ and $F\cap {\mathcal{F}}_2\ne \varnothing$, i.e., there is some $f_i\in {\mathcal{F}}_2$ for $i\in [\Delta ]$. It suffices to consider all situations in Observation 1. Note that v sends $\frac{3}{2}$ to each $f\in {\mathcal{F}}_2\cap F$.

(s1) (see in Figure 6a): As every vertex $u_i$ that is not incident to faces in ${\mathcal{F}}_2\cap F$ is $3^{+}$-vertex, v sends at most $\frac{3}{2}$ to the possible 3-face and at most 1 to every $4^{+}$-faces not contained in the petal. Thus, $c{h}^{*}\left(v\right)\ge 2\Delta -6-(4+1)\times \frac{3}{2}-1\times (\Delta -5)\ge 0$ since $\Delta \ge 9$.

(s2) Assume v has a petal $P=\{f_1,f_2,f_3\}$ (as shown in Figure 6b). By (e) in Lemma 8, the degree of $u_i$ is at least 3 for $i\in [4,\Delta ]$. If $|f_{\Delta }|=3$, then $f_1$ has a chord $v{u}_{\Delta }$. So that is impossible and $f_{\Delta }$ must be a $4^{+}$-face. Therefore, v sends at most $\frac{3}{2}$ to the possible 3-face, at most $\frac{5}{4}$ to $f_{\Delta }$, at most 1 to other $4^{+}$-faces, and so $c{h}^{*}\left(v\right)\ge 2\Delta -6-(3+1)\times \frac{3}{2}-\frac{5}{4}-1\times (\Delta -5)>0$ since $\Delta \ge 9$.

(s3) Assume v has exactly one petal $P=\{f_1,f_2\}$. Similarly, v sends $\frac{3}{2}$ to $f_i$ for $i\in \left[2\right]$ and the degree of $u_i$ is at least 3, for $i\in [4,\Delta ]$. If one of $f_3$ and $f_{\Delta }$ is a 3-face, say $f_3$, then $f_2$ as a face has a chord $v{u}_4$, a contradiction. So $f_3$ and $f_{\Delta }$ are $4^{+}$-face. Therefore, v sends at most $\frac{5}{4}$ to $f_3,f_{\Delta }$, sends at most $\frac{3}{2}$ to the possible 3-face and sends at most 1 to other $4^{+}$-faces. So, $c{h}^{*}\left(v\right)\ge 2\Delta -6-2\times \frac{3}{2}-2\times \frac{5}{4}-\frac{3}{2}-1\times (\Delta -5)\ge 0$ since $\Delta \ge 9$.

Before the analysis of (s4) and (s5), let ${\mathcal{P}}_i$ be the set of all petals of v with size i and $|{\mathcal{P}}_i|={\omega }_i$ for $i\in \left[2\right]$. Set $\mathcal{P}={\mathcal{P}}_1\cup {\mathcal{P}}_2$. Then, $\left|\mathcal{P}\right|={\omega }_1+{\omega }_2=\omega$. Let $F^{\prime }=\{f_i\in F:f_i\notin P\in \mathcal{P}\mathrm{and}{f}_i\mathrm{shares}\mathrm{an}(\Delta ,3)-\mathrm{edge}\mathrm{with}\mathrm{a}\mathrm{face}\mathrm{in}F\cap {\mathcal{F}}_2\}$ and set $|F^{\prime }|=\alpha$. Note that for any $f\in F$:

• If $f\in F^{\prime }$ and $\left|f\right|\ge 4$, then v sends at most 1 to f.
• If f ∈ F^′ and |f| = 3, then v sends $\frac{3}{2}$ to f.
• If $f\in P$ for some $P\in \mathcal{P}$, then v sends $\frac{3}{2}$ to f.
• Otherwise, v sends at most $\frac{5}{4}$ to every $4^{+}$-face f.

Now we consider two cases based on the existence of a 3-face.

Case 1.

For any $i\in [\Delta ]$, $|f_i|\ge 4$.

(s4) Note that ${\omega }_2\ne 0$ and each $P\in {\mathcal{P}}_2$ is sharing $(\Delta ,3)$-edges with two faces in $F^{\prime }$. When $\alpha \le {\omega }_2$, the only possibility is that $\alpha ={\omega }_2$ and ${\omega }_1=0$. In this situation, every face in $F^{\prime }$ shares $(\Delta ,3)$-edge with two different petals in ${\mathcal{P}}_2$ respectively. Hence, $\alpha ={\omega }_2=\frac{\Delta }{3}$ implies that $c{h}^{*}\left(v\right)=2\Delta -6-3{\omega }_2-\frac{\Delta }{3}=\frac{2}{3}\Delta -6\ge 0$ by $\Delta \ge 9$.

Thus, $\alpha$ is at least ${\omega }_2+1$. So,

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-\frac{3}{2}{\omega }_1-3{\omega }_2-({\omega }_2+1)-\frac{5}{4}(\Delta -{\omega }_1-2{\omega }_2-({\omega }_2+1))\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{23}{4}-\frac{1}{4}{\omega }_2-\frac{1}{4}{\omega }_1\ge 0,\hfill \end{array}$$

since $\Delta \ge 9$ and ${\omega }_1+{\omega }_2=\omega \le ⌊\frac{\Delta }{2}⌋$.

(s5) Note that ${\omega }_1\ne 0,{\omega }_2=0$. As each $P\in \mathcal{P}$ is sharing an $(\Delta ,3)$-edge with some face in $F^{\prime }$, $\alpha \ge ⌈\frac{{\omega }_1}{2}⌉$. Thus, we have

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-\frac{3}{2}{\omega }_1-⌈\frac{{\omega }_1}{2}⌉-\frac{5}{4}(\Delta -{\omega }_1-⌈\frac{{\omega }_1}{2}⌉)\hfill \\ \hfill & =\frac{3}{4}\Delta -6-\frac{1}{4}{\omega }_1+\frac{1}{4}⌈\frac{{\omega }_1}{2}⌉\ge 0,\hfill \end{array}$$

since $\Delta \ge 9$ and ${\omega }_1=\omega \le ⌊\frac{\Delta }{2}⌋$.

Case 2.

There is some $i\in [\Delta ]$ such that $f_i$ is a 3-face. We may assume that $|f_1|=3$ and $d\left(u_1\right)\le d\left(u_2\right)$.

(s4) Similarly, we have $\alpha$ is at least ${\omega }_2+1$ and ${\omega }_2\ne 0$. Note that v sends at most $\frac{3}{2}$ to each petal in ${\mathcal{P}}_1$ and sends at most 3 to each petal in ${\mathcal{P}}_2$. Assume that $f_1$ is not a $(2,\Delta ,\Delta )$-face. If $f_1\notin F^{\prime }$, then v sends at most $\frac{3}{2}$ to $f_1$, sends at most 1 to $w_2+1$ faces in $F^{\prime }$, and sends at most $\frac{5}{4}$ to other $4^{+}$-face. Thus, we have

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-\frac{3}{2}-\frac{3}{2}{\omega }_1-3{\omega }_2-({\omega }_2+1)-\frac{5}{4}(\Delta -1-{\omega }_1-2{\omega }_2-{\omega }_2-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{24}{4}-\frac{1}{4}{\omega }_1-\frac{1}{4}{\omega }_2\hfill \end{array}$$

(1)

If $f_1\in F^{\prime }$, then $f_1$ contains a $(\Delta ,3)$-edge and one of $f_2,f_{\Delta }\in {\mathcal{F}}_2$, say $f_{\Delta }\in {\mathcal{F}}_2$. Thus, $f_2$ is a $4^{+}$-face containing a $(\Delta ,{(\Delta -1)}^{+})$-edge. If $f_2\notin F^{\prime }$, then v sends at most $\frac{3}{2}$ to $f_1$, sends at most 1 to $f_2$ and $w_2$ faces in $F^{\prime }$. Thus, we have

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-\frac{3}{2}-\frac{3}{2}{\omega }_1-3{\omega }_2-{\omega }_2-1-\frac{5}{4}(\Delta -1-{\omega }_1-2{\omega }_2-{\omega }_2-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{24}{4}-\frac{1}{4}{\omega }_1-\frac{1}{4}{\omega }_2.\hfill \end{array}$$

(2)

Otherwise, $f_2\in F^{\prime }$ and $f_2$ is a $4^{+}$-face containing a $(\Delta ,3)$-edge. Thus, $f_2$ receives at most $\frac{3}{4}$ from v by (R2), (R4) and (R5). And so,

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-\frac{3}{2}-\frac{3}{2}{\omega }_1-3{\omega }_2-({\omega }_2-1)-\frac{3}{4}-\frac{5}{4}(\Delta -1-{\omega }_1-2{\omega }_2-({\omega }_2-1)-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{24}{4}-\frac{1}{4}{\omega }_1-\frac{1}{4}{\omega }_2.\hfill \end{array}$$

(3)

As $\Delta \ge 9$ and ${\omega }_1+{\omega }_2=\omega \le ⌊\frac{\Delta }{2}⌋$, Equations (1)–(3) are not negative except when $\Delta =9$ and ${\omega }_2=1,{\omega }_1=3$. Then, $f_1$ is adjacent to some $P\in {\mathcal{P}}_1$ and some $P\in {\mathcal{P}}_2$, and so there is a $(2,3)$-edge or $(3,3)$-edge in $f_1$. It is a contradiction since $d\left(u\right)+d\left(v\right)\ge \Delta +2$ for any $uv\in E\left(G\right)$.

Assume that $f_1$ is a $(2,\Delta ,\Delta )$-face. Clearly, we have $f_1\notin F^{\prime }$ and one of $f_2,f_{\Delta }$ contains a $(\Delta ,\Delta )$-edge. Without loss of generality, we may assume that $f_2$ is a $4^{+}$-face with a $(\Delta ,\Delta )$-edge. Similarly, if $f_2\notin F^{\prime }$, then $f_2$ receives at most 1 from v by (R2), (R4) and (R5). Thus,

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-2-\frac{3}{2}{\omega }_1-3{\omega }_2-({\omega }_2+1)-1-\frac{5}{4}(\Delta -1-{\omega }_1-2{\omega }_2-({\omega }_2+1)-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{25}{4}-\frac{1}{4}{\omega }_1-\frac{1}{4}{\omega }_2.\hfill \end{array}$$

(4)

Otherwise, $f_2\in F^{\prime }$ and $f_2$ is a $4^{+}$-face containing a $(\Delta ,3)$-edge. Thus, each $f\in F^{\prime }-f_2$ receives at most 1 from v and $f_2$ receives at most $\frac{3}{4}$ from v by (R2), (R4) and (R5). So,

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-2-\frac{3}{2}{\omega }_1-3{\omega }_2-{\omega }_2-\frac{3}{4}-\frac{5}{4}(\Delta -1-{\omega }_1-2{\omega }_2-{\omega }_2-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{25}{4}-\frac{1}{4}{\omega }_1-\frac{1}{4}{\omega }_2.\hfill \end{array}$$

(5)

As $\Delta \ge 9$, ${\omega }_1+{\omega }_2=\omega \le ⌊\frac{\Delta }{2}⌋$ and the existence of the 3-face, Equations (4) and (5) are not negative unless when $\Delta =9$ and ${\omega }_2=2,{\omega }_1=1$, or ${\omega }_2=1,{\omega }_1=2$. Note that v sends 2 to face $f_1$, sends 3 to each petal in ${\mathcal{P}}_2$ and sends $\frac{3}{2}$ to each petal in ${\mathcal{P}}_1$.

When $\Delta =9$ and ${\omega }_2=2,{\omega }_1=1$, it suffices to consider the structure in Figure 7a by symmetry and the existence of $(2,\Delta ,\Delta )$-face. Note that $f_5$ and $f_8$ are not 4-faces due to (f) in Lemma 8. So, $f_5$ and $f_8$ receive at most $\frac{1}{3}$ from v by (R5.2) respectively. Thus, we have $c{h}^{*}\left(v\right)\ge 12-6-\frac{3}{2}-2-1-\frac{1}{3}\times 2\ge 0$.

Figure 7. When $\Delta =9$ and ${\omega }_2=2,{\omega }_1=1$, or ${\omega }_2=1,{\omega }_1=2$.

Now, we consider the situation when $\Delta =9$ and ${\omega }_2=1,{\omega }_1=2$. Note that for any $4^{+}$-face $f\in F^{\prime }$, f receives at most 1 from v. By all degree conditions and the absence of chord in any face, the unique $(2,\Delta ,\Delta )$-face does not share any edge with any petals. All situations are shown in Figure 7b–h. Based on $F\cap {\mathcal{F}}_1=\varnothing$ and (R4.1, R5.1). Thus, we have $c{h}^{*}\left(v\right)\ge 12-6-2-4*1=0$.

(s5) Note that ${\omega }_1=\omega \le ⌊\frac{\Delta }{2}⌋$ and $\alpha \ge ⌈\frac{{\omega }_1}{2}⌉$.

Assume $f_1$ is not a $(2,\Delta ,\Delta )$-face. If $f_1\notin F^{\prime }$, then

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-\frac{3}{2}-\frac{3}{2}{\omega }_1-⌈\frac{{\omega }_1}{2}⌉-\frac{5}{4}(\Delta -1-{\omega }_1-⌈\frac{{\omega }_1}{2}⌉)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{25}{4}-\frac{1}{4}{\omega }_1+\frac{1}{4}⌈\frac{{\omega }_1}{2}⌉\ge 0,\hfill \end{array}$$

since $\Delta \ge 9$.

If $f_1\in F^{\prime }$, then we can assume that $f_{\Delta }\in {\mathcal{F}}_2$, and so $f_2$ is a $4^{+}$-face containing a $(\Delta ,{(\Delta -1)}^{+})$-edge. Note that $|F^{\prime }-f_1|\ge ⌈\frac{{\omega }_1-1}{2}⌉$. If $f_2\notin F^{\prime }$, then $f_2$ receives at most 1 from v by (R2), (R4) and (R5). Thus,

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-\frac{3}{2}-\frac{3}{2}{\omega }_1-⌈\frac{{\omega }_1-1}{2}⌉-1-\frac{5}{4}(\Delta -1-{\omega }_1-⌈\frac{{\omega }_1-1}{2}⌉-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{24}{4}-\frac{1}{4}{\omega }_1+\frac{1}{4}⌈\frac{{\omega }_1-1}{2}⌉\ge 0,\hfill \end{array}$$

since $\Delta \ge 9$.

Otherwise, $f_2\in F^{\prime }$ and $f_2$ is a $4^{+}$-face containing a $(\Delta ,3)$-edge. Then, $|F^{\prime }-\{f_1,f_2\}|\ge ⌈\frac{{\omega }_1-2}{2}⌉$ and each $f\in F^{\prime }-\{f_1,f_2\}$ receives at most 1 from v and $f_2$ receives at most $\frac{3}{4}$ from v by (R2), (R4), and (R5). So,

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-\frac{3}{2}-\frac{3}{2}{\omega }_1-⌈\frac{{\omega }_1-2}{2}⌉-\frac{3}{4}-\frac{5}{4}(\Delta -1-{\omega }_1-⌈\frac{{\omega }_1-2}{2}⌉-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{23}{4}-\frac{1}{4}{\omega }_1+\frac{1}{4}⌈\frac{{\omega }_1-2}{2}⌉\ge 0,\hfill \end{array}$$

since $\Delta \ge 9$.

Assume that $f_1$ is a $(2,\Delta ,\Delta )$-face. Clearly, we have $f_1\notin F^{\prime }$. Similarly, if $f_2\notin F^{\prime }$, then $f_2$ receives at most 1 from v by (R4) and (R5). Thus,

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-2-\frac{3}{2}{\omega }_1-⌈\frac{{\omega }_1}{2}⌉-1-\frac{5}{4}(\Delta -1-{\omega }_1-⌈\frac{{\omega }_1}{2}⌉-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{26}{4}-\frac{1}{4}{\omega }_1+\frac{1}{4}⌈\frac{{\omega }_1}{2}⌉.\hfill \end{array}$$

(6)

Otherwise, $f_2\in F^{\prime }$ and $f_2$ is a $4^{+}$-face containing a $(\Delta ,3)$-edge. Thus, each $f\in F^{\prime }-f_2$ receives at most 1 from v and $f_2$ receives at most $\frac{3}{4}$ from v by (R2), (R4) and (R5). Thus,

$$\begin{array}{cc}\hfill c{h}^{*}\left(v\right)& \ge 2\Delta -6-2-\frac{3}{2}{\omega }_1-⌈\frac{{\omega }_1-1}{2}⌉-\frac{3}{4}-\frac{5}{4}(\Delta -1-{\omega }_1-⌈\frac{{\omega }_1-1}{2}⌉-1)\hfill \\ \hfill & =\frac{3}{4}\Delta -\frac{25}{4}-\frac{1}{4}{\omega }_1+\frac{1}{4}⌈\frac{{\omega }_1-1}{2}⌉.\hfill \end{array}$$

(7)

Combined with $\Delta \ge 9$ and ${\omega }_1\le ⌊\frac{\Delta }{2}⌋$, Equations (6) and (7) are not negative unless $\Delta =9$, ${\omega }_1=4$ and $\alpha =2$ (shown in Figure 8). Similarly, by all degree conditions and the absence of chord of any face, the unique 3-face $f_1$ does not appear in any position. □

Figure 8. $\Delta =9$, ${\omega }_1=4$ and $\alpha =2$.

By Claims 1 and s, for $d\in [3,\Delta ]$, the final charge of d-vertex is nonnegative. The proof of Theorem 7 is completed.

5. Conclusions

Theorems 6 and 7 are two main results in our paper, we study the edge DP-chromatic number of $K_4$-minor free graphs and planar graph with $\Delta \ge 9$, respectively. Moreover, we also prove that the upper bound in Theorem 6 is sharp by one example.

For $K_4$-minor free graphs. On the one hand, by Theorem 6, the corresponding edge DP-chromatic number is $\Delta$ or $\Delta +1$. On the other hand, we give one example to demonstrate that there exists some $K_4$-minor free graph satisfying with the edge DP-chromatic number is not $\Delta$. It reflects that the upper bound in Theorem 6 is sharp.

For a planar graph G with $\Delta \ge 9$, some researchers proved ${\chi }_{DP}^{\prime }\left(G\right)\le \Delta +1$. One sufficient condition is given for G such that ${\chi }_{DP}^{\prime }\left(G\right)=\Delta$ in Theorem 7.