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Research Article | Open Access

Volume 2021 |Article ID 3562513 | https://doi.org/10.1155/2021/3562513

Wenwen Zhang, "Planar Graphs of Maximum Degree 6 and without Adjacent 8-Cycles Are 6-Edge-Colorable", Journal of Mathematics, vol. 2021, Article ID 3562513, 10 pages, 2021. https://doi.org/10.1155/2021/3562513

# Planar Graphs of Maximum Degree 6 and without Adjacent 8-Cycles Are 6-Edge-Colorable

Academic Editor: M. M. Bhatti
Accepted21 May 2021
Published01 Jun 2021

#### Abstract

In this paper, by applying the discharging method, we show that if is a planar graph with a maximum degree of that does not contain any adjacent 8-cycles, then is of class 1.

#### 1. Introduction

Graph coloring is a very important problem in graph theory. Since the four-color problem was first proposed, many other forms of coloring problems have been put forward and extensively studied. In this paper, we study the edge coloring of a planar graph. It is assumed that all graphs are finite, without parallel edges, and all edges are undirected. A planar graph is a graph that can be embedded in the plane satisfying the condition that no two edges intersect geometrically except at a vertex to which they are both incident. Table 1 presents some symbols and definitions used in this paper.

 The set of all edges incident with Cardinality of Neighbor set of -, -, -vertex Vertex with , , or -, -neighbor of a vertex - or -vertex adjacent to -cycle Cycle of length The set of all faces of a planar graph Number of edges on with each cut edge calculated twice -, -face Face with , Number of -faces incident with Number of -neighbors of i-th smallest degree of a neighbor of Length of the shortest cycle of a graph

A graph is said to be -edge-colorable if the edges of can be colored with colors such that adjacent edges have different colors. In all edge colorings of , the edge chromatic number of a graph is the minimum number of colors used. In 1964, Vizing proved that, for any graph of maximum degree , or . This result divides all graphs into two classes: graphs satisfying are in class 1, whereas graphs satisfying are in class 2. For a connected graph , if it is of class 2 and for , then is critical. A -critical graph is a critical graph of maximum degree .

For planar graphs, Vizing [1] proved that any planar graph with is of class 1 and illustrated that there are class 2 planar graphs with . Therefore, Vizing conjectured that any planar graph with is of class 1. Zhang [2] and Sanders and Zhao [3] proved the case of independently. The case of has not yet been solved, but some positive results with restrictive conditions have been obtained, as in Theorem 1.

Theorem 1. If is one of the following planar graphs, then is of class 1:(1) and , or and , or and ([1])(2) without intersecting 3-cycles ([4])(3) without 4- or 5-cycles ([5])(4) without 5- or 6-cycles with chords [6](5) without 5-cycles ([7]) or 6-cycles ([8]) with two chords(6) without 7-cycles ([9]) or 6-cycles ([10]) with three chords(7) and any vertex is incident with at most three triangles ([11])(8) and without adjacent -cycles, where ([12])(9) and any two 7-cycles are not adjacent ([13]).

In this paper, two cycles sharing at least a common edge are said to be adjacent. We present a result concerning the edge chromatic number of planar graphs with which do not contain any adjacent 8-cycles.

#### 2. Main Result and Its Proof

To prove our main result, we first present several lemmas.

Lemma 1 (see [2, 3]). Suppose that is a planar graph with . Then, .

Lemma 2 (Vizingâ€™s Adjacency Lemma [1]). Let be a -critical graph, and let with .(1)If , then has at least -neighbors(2)If , then has at least two -neighbors

Lemma 3 (see [2]). Let be a -critical graph, and let with . Then,(1)Every vertex of is a -vertex(2)Every vertex of has a degree of at least (3)If , then every vertex of is a -vertex

Lemma 4 (see [3]). No -critical graph has distinct vertices , , and such that is adjacent to and , , and is in at least triangles not containing .

We use to denote a 3-face with three vertices satisfying . A face is denoted by a bad 4-face in the event that and form a 3-face. Let denote the number of bad 4-faces incident with such that is incident with a bad 4-face with . Let , so that .

Lemma 5. Let be a 6-critical plane graph with no two adjacent 8-cycles. Let .(1)If , and , then the remaining face is a -face. Moreover, if , then there is at least one 5-neighbor of on the -face and .(2)If and , then the remaining face is an -face or a 4-face (as in case 1.1 of Figure 1). Moreover,â€‰(2.1) if the remaining face is an -face and any 3-neighbors and 5-neighbors are not adjacent, then for each 5-neighbor of â€‰(2.2) if the remaining face is a 4-face and , then for each - neighbor of â€‰(2.3) if the remaining face is a 4-face and , then there are two cases for each 6-neighbor of â€‰(2.3.1) if , then , and â€‰(2.3.2) if , then and (3)If and , then the remaining face is an -face other than cases 1.2â€“1.7 in Figure 1.â€‰(3.1) and in Figure 1 (cases 1.2, 1.4, and 1.5)â€‰(3.2) , and in Figure 1 (case 1.3)â€‰(3.3) and in Figure 1 (cases 1.6 and 1.7)(4)If and , then , , or (as in Figure 1, case 1.8). Furthermore, if , then there are three cases for each 6-neighbor of â€‰(4.1) if , then , and â€‰(4.2) if , then and â€‰(4.3) if , then , and (5)If is on a bad 4-face satisfying and the remaining face is a 4-face or 5-face (as in Figure 1, cases 1.9â€“1.25), then there are four cases for vertex :â€‰(5.1) , and in Figure 1, cases 1.9â€“1.15â€‰(5.2) , and in Figure 1, cases 1.16â€“1.21â€‰(5.3) and in Figure 1, cases 1.22â€“1.24â€‰(5.4) , and in Figure 1, case 1.25(6)If , and is on a 3-face satisfying and another face incident with is a 5-face (as in Figure 1, cases 1.26â€“1.30), then there are three cases for vertex :â€‰(6.1) , and in Figure 1, case 1.26â€‰(6.2) , and in Figure 1, case 1.27â€‰(6.3) , and in Figure 1, cases 1.28â€“1.30