List Edge Colorings of Planar Graphs without Adjacent 7-Cycles

Edge coloring and list edge coloring of graphs are very old fashioned problems in graph theory, and the research on such problems has a long history. Denote Z as the set of the integers. Now, we only consider the list edge coloring problem of finite simple undirected graphs. Before describing the concept of list edge coloring in detail, we have to revisit fundamental conception of normal edge coloring. A graph G is k-edge-colorable if all edges of G can be colored by k colors such that no two adjacent edges get the same colors. Denoted χ′(G) as the edge chromatic number of a graph G, which is the smallest k ∈ Z satisfying G is k-edgecolorable. For each edge e of graph G, if we can assign a list L(e) of colors to it, then L is an edge assignment. Graph G is edge-L-colorable if G has a proper edge-coloring φ such that φ(e) ∈ L(e) for each edge e of G, and φ is an edge-L-coloring. Graph G is edge-k-choosable if for every L satisfying |L(e)|≥ k(k ∈ Z) for each edge e. Denote χl ′(G) as the listedge chromatic number of G which is the smallest k in Z such that G is edge-k-choosable. We will study the list edge colorings of planar graphs. Planar graph is a kind of graph broadly studied in graph coloring theory.-e so-called plane graph is actually a special drawing method of planar graph, which can be embedded in the plane satisfying no two edges intersect geometrically except at a vertex to which they are both incident. Let us introduce some definitions and symbols needed. Given a plane graph G, we use V(G), E(G), F(G),Δ(G), and δ(G) to indicate its vertex set, edge set, face set, maximum degree, and minimum degree, respectively. For a vertex v ∈ V(G), let EG(v) or E(v) be the set of edges which are incident with v. We use dG(v) or d(v) to indicate the degree of v inG, which is the number of edges in E(v). We use NG(v) or N(v) to indicate the set of the vertices which are adjacent to v in G. Denoted k-vertex, k -vertex, or k-vertex as a vertex of degree k, at most k or at least k, respectively. A k (or k)-neighbor of x is a k (or k)-vertex which is adjacent to a vertex x. A k-cycle is a cycle of length k. Two cycles are adjacent, that is, the two cycles share at least a common edge. A 2-alternating cycle is an even cycle in which the 2-vertices appear alternately. For f ∈ F(G), we use dG(f) to indicate the degree of a face, which is the number of edges incident with f where each cut edge is counted twice. Denote k-, k-face as a face of degree k, at least k. For a k-face of G, we called it (i1, i2, . . . , ik)-face if the vertices incident with it are of degrees i1, i2, . . . , ik, respectively. We use fk(v) to indicate the number of k-faces which are incident with v, dk(f) the number of k-vertices which are incident withf, and dk(v) the number of k-vertices which are incident with v.


Introduction
Edge coloring and list edge coloring of graphs are very old fashioned problems in graph theory, and the research on such problems has a long history. Denote Z + as the set of the integers. Now, we only consider the list edge coloring problem of finite simple undirected graphs. Before describing the concept of list edge coloring in detail, we have to revisit fundamental conception of normal edge coloring. A graph G is k-edge-colorable if all edges of G can be colored by k colors such that no two adjacent edges get the same colors. Denoted χ ′ (G) as the edge chromatic number of a graph G, which is the smallest k ∈ Z + satisfying G is k-edgecolorable. For each edge e of graph G, if we can assign a list L(e) of colors to it, then L is an edge assignment. Graph G is edge-L-colorable if G has a proper edge-coloring ϕ such that ϕ(e) ∈ L(e) for each edge e of G, and ϕ is an edge-L-coloring. Graph G is edge-k-choosable if for every L satisfying |L(e)| ≥ k(k ∈ Z + ) for each edge e. Denote χ l ′ (G) as the listedge chromatic number of G which is the smallest k in Z + such that G is edge-k-choosable.
We will study the list edge colorings of planar graphs. Planar graph is a kind of graph broadly studied in graph coloring theory. e so-called plane graph is actually a special drawing method of planar graph, which can be embedded in the plane satisfying no two edges intersect geometrically except at a vertex to which they are both incident. Let us introduce some definitions and symbols needed. Given a plane graph G, we use V(G), E(G), F(G), Δ(G), and δ(G) to indicate its vertex set, edge set, face set, maximum degree, and minimum degree, respectively. For a vertex v ∈ V(G), let to indicate the degree of v in G, which is the number of edges in E(v). We use N G (v) or N(v) to indicate the set of the vertices which are adjacent to v in G. Denoted k-vertex, k − -vertex, or k + -vertex as a vertex of degree k, at most k or at least k, respectively. A k (or k + )-neighbor of x is a k (or k + )-vertex which is adjacent to a vertex x. A k-cycle is a cycle of length k. Two cycles are adjacent, that is, the two cycles share at least a common edge. A 2-alternating cycle is an even cycle in which the 2-vertices appear alternately. For f ∈ F(G), we use d G (f) to indicate the degree of a face, which is the number of edges incident with f where each cut edge is counted twice. Denote k-, k + -face as a face of degree k, at least k. For a k-face of G, we called it (i 1 , i 2 , . . . , i k )-face if the vertices incident with it are of degrees i 1 , i 2 , . . . , i k , respectively. We use f k (v) to indicate the number of k-faces which are incident with v, d k (f) the number of k-vertices which are incident with f, and d k (v) the number of k-vertices which are incident with v.
Wang and Wu [13] proved that G is edge-k-choosable (k � max 10, Δ(G) { }) for a planar graph G without 6-cycles with three chords. Now, we will study the planar graphs without adjacent 7-cycles and obtain: Theorem 1. Suppose that Gis a planar graph which contains no adjacent 7-cycles.
From eorem 1, we can obtain the following corollary.

The Proofs of Theorem
) be a minimal graph satisfying the number of |E(G)| as little as possible; then, the graph G has the following properties.
□ Lemma 1 (see [14]). Let Gbe a planar graph, by the minimality hypothesis of graphG, and we have (2) and (3), G 2 contains no odd cycle and even cycle. erefore, G 2 must be a forest. ereby, there must be a matching M in G 2 and all 2-vertices in M are saturated. If v 1 v 2 ∈ M and d G 2 (v 1 ) � 2, then v 2 is named the 2-master of v 1 and v 1 is the dependent of v 2 . Obviously, each 2-vertex has a 2-master and each k-vertex may be the 2-master of no more than one 2vertex.
Note that, in Lemma 2, we mark y as the 3-master of x if xy ∈ M ′ and x ∈ X.

Lemma 3
Suppose that G is a planar graph which contains no adjacent 7-cycles and d(v) � 9. en, (1) If f 3 (v) � 6 and one of its edges is not incident with any 3-face (as in Figure 1, (1) Now, we prove (1-10) in Figure 1. Suppose that f 1 , f 2 , . . . , f 5 and f 7 are 3-faces. If f 6 is a 4-face or 5-face or 6-face, then there will be adjacent 7-cycles in G. It must be d(f 6 ) ≥ 7 and f 7 + (v) ≥ 1. e proof process of (1-11) and (1-13) is similar, so we will not repeat it. (2) Its proof process is similar to (1), so we will not repeat it.
Similarly, we can get the following two lemmas.  Figure 2(2 − 1)), then  43)), then figure (3 − 47)), then f 7 + (v) � 2 Now, we complete the proof by Euler's formula. In [6], the authors proved χ l ′ (G) � Δ(G) for every planar graph with Δ(G) ≥ 12, so we only assume that Δ(G) ≤ 11 in our proof. Suppose that G is already embedded in the plane. We can obtain by practical Euler's formula |V(G)| − |E(G)| + |F(G)| � 2. Firstly, denote ch as the original charge. For each Secondly, we formulate some rules to redistribute the original charge and each x ∈ V ∪ F will get a new charge ch ′ (x). Note that the rules we formulated only move between the vertices and faces of the graph and have no effect on the total charge. irdly, we will show that ch ′ (x) ≥ 0 (x ∈ V ∪ F). If we can do, then we will obtain an apparent contradiction 0 ≤ ch ′ (x) � ch(x) < 0 (x ∈ V ∪ F). e proof of eorem 1 is completed.
The discharging rules are formulated as in R1-R5. We use c(x ⟶ y) to indicate the charge from x to y:

R3.
Every 5-vertex receives (2/15) from each of its 7 + -neighbors. 1, 2, 3).    Now, let us start to test and verify ch ′ is greater than or equal to 0 for all vertices and faces. It is easy to verify faces, so let us verify the new charge of every face firstly. Obviously, Let us verify the new charge for every vertex.