A Note on Edge-Group Choosability of Planar Graphs without 5-Cycles

Abstract

This paper is devoted to a study of the concept of edge-group choosability of graphs. We say that is edge--group choosable if its line graph is -group choosable. In this paper, we study an edge-group choosability version of Vizing conjecture for planar graphs without 5-cycles and for planar graphs without noninduced 5-cycles (2010 Mathematics Subject Classification: 05C15, 05C20).

1. Introduction

We consider only simple graphs in this paper unless otherwise stated. For a graph , we denote its vertex set, edge set, minimum degree, and maximum degree by , , , and , respectively. A plane graph is a particular drawing of a planar graph in the Euclidean plane. We denote the set of faces of a plane graph by . For a plane graph and , we write if are the vertices on the boundary walk of enumerated clockwise. The degree of a face is the number of edges on the boundary walk. Let , or simply , denote the degree of a vertex (or face) in . A vertex (or face) of degree is called a -vertex (or -face). For , is the set of all vertices of that are adjacent to in . We denote the line graph of a graph by .

A -coloring of a graph is a mapping from to the set of colors such that for every edge . A graph is -colorable if it has a -coloring. The chromatic number is the smallest integer such that is -colorable. A mapping is said to be a list assignment for if it supplies a list of possible colors to each vertex . A -list assignment of is a list assignment with for each vertex . If has some -coloring such that for each vertex , then is -colorable or is an -coloring of . We say that is -choosable if it is -colorable for every -list assignment. The choice number or list chromatic number is the smallest such that is -choosable. For edge-colorings of , we can define analogous notions such as edge--colorability, edge--choosability, the chromatic index, and the choice index. Clearly, we have and . The notion of list coloring of graphs has been introduced by Erdős et al. [1] and Vizing [2]. The following conjecture, which first appeared in [3], is well-known as the List Edge Coloring Conjecture.

Conjecture 1. If is a multigraph, then .

Although Conjecture 1 has been proved for a few special cases such as bipartite multigraphs, complete graphs of odd order, multicircuits, graphs with that can be embedded in a surface of nonnegative characteristic, and outerplanar graphs, it is regarded as very difficult. Vizing proposed the following weaker conjecture (see [4]).

Conjecture 2. Every graph is edge--choosable.

Assume is an Abelian group, and denotes the set of all functions . Consider an arbitrary orientation of . The graph is -colorable if, for every , there is a vertex coloring such that for each directed edge from to . The group chromatic number of , , is the minimum such that is -colorable for any Abelian group of order at least . The notion of group coloring of graphs was first introduced by Jaeger et al. [5].

The concept of group choosability was introduced by Král and Nejedlý in [6], and some first results in this area were obtained in [7, 8]. Let be an Abelian group of order at least and be a list assignment of . For , an -coloring under an orientation of is an -coloring such that for every edge , where is directed from to . If for each , there exists an -coloring for , and then we say that is -colorable. The graph is -group choosable if is -colorable for each Abelian group of order at least and any -list assignment. The minimum for which is -group choosable is called the group choice number of and is denoted by . It is clear that the concept of group choosability is independent of the orientation on . Graph is called edge--group choosable if its line graph is -group choosable. The group-choice index of , , is the smallest such that is edge--group choosable, i.e., . It is easily seen that an even cycle is not edge-2-group choosable. This example shows that is not generally equal to . But we can extend the Vizing conjecture as follows.

Conjecture 3. If is a multigraph, then .

Since , as a sufficient condition, we have the following weaker conjecture.

Conjecture 4. If is a multigraph, then .

Some early results concerning edge-group choosability of graphs were presented by the authors in a series of lectures in Annual Iranian Mathematical Conferences (see [9–11]). Conjecture 3 has been proved for graphs with maximum degree [9], planar graphs with maximum degree [9], planar graphs without 4-cycles with maximum degree [11], outerplanar graphs [12], simple series-parallel graphs [12], -minor-free graphs [12], and planar graphs with maximum degree that has no cycles of length from 4 to 14 [10]. For further reference, we add here some related details.

Theorem 1 (see [9]). Let be a natural number, be a vertex of degree at most 2 of , and be an edge incident to . If , then .

Theorem 2 (see [9]). Let be a graph with , for each . Then, .

Theorem 3 (see [9]). Let be a graph with maximum degree . If , then , and if , then .

Theorem 4 (see [7]). (a)Let and denote a path and a cycle of length , respectively. Then, and .(b)For any connected simple graph , we have , with equality holds if and only if is either a cycle or a complete graph.

Immediately from Theorem 4, we see that and . In this paper, we show that any planar graph without 5-cycles with maximum degree is edge--group choosable. If in addition , we can show that is edge--group choosable. This proves in advance that Conjecture 3 and, consequently, Conjecture 4 holds for this class of planar graphs. Moreover, we show that if is a planar graph without noninduced 5-cycles, then .

2. Main Results

First we need a lemma, which we will discuss below. It is a structural lemma for plane graphs without 5-cycles.

Lemma 1 (see [13]). If a plane graph with has no five cycles, then there exists an edge of such that and .

Note that is a minimal counterexample to a theorem if is a counterexample, that is, satisfies the hypotheses but not the conclusion of the theorem, and there is no counterexample satisfying either or and .

Theorem 5. If is a planar graph without 5-cycles with maximum degree , then is edge--group choosable.

Proof. We saw in Theorem 4 that if and denote a path and a cycle of length , respectively, then and . Moreover, for any connected simple graph , we have , with equality holds if and only if is either a cycle or a complete graph. Immediately, we see that and . Hence, if , then , and if , then . Here, we used the observation that, for a connected graph , if , then ; if , then or ; if , then ; and if , then . Now, let be a minimal counterexample to Theorem 5 for some Abelian group with , a -list assignment and . Then, is connected, , and . By Lemma 1, there exists a vertex with . Suppose that . Then, , and since , there exists an -coloring . For each we can consider, without loss of generality, to be directed from to . Then, since and , . In other words, there is at least one color available to color . Thus, we can color all edges of . This contradiction completes the proof of theorem.

The above proof shows that the only critical case is . If remove it, we can prove a stronger result.

Theorem 6. If is a planar graph with maximum degree and without 5-cycles, then .

Proof. Let be a minimal counterexample to this theorem for some Abelian group with , a -list assignment and . Then, . By Lemma 1, there exists a vertex with . Suppose that . Then, , and since , there exists an -coloring . For each , we can consider, without loss of generality, to be directed from to . Then, since and , . In other words, there is at least one color available to color . Thus, we can color all edges of . This contradiction completes the proof of theorem.

The structure of planar graphs without noninduced 5-cycles is given in the following lemma.

Lemma 2 (see [14]). Let be a planar graph without noninduced 5-cycles. Then, contains one of the following configurations:(1)An edge with (2)An even cycle with and

Theorem 7. If is a planar graph without noninduced 5-cycles, then

Proof. Using Lemma 2, the proof is straightforward and is similar to the proof of Theorem 5. We leave the details to the reader.

If a planar graph without noninduced 5-cycles in addition contains no even cycles, we can replace by in Theorem 7.

Theorem 8. If is a planar graph without noninduced 5-cycles and without even cycle with and , then .

Proof. Let and be a minimal counterexample to this theorem for some Abelian group with , a -list assignment and . By Theorem 1 and Lemma 2, there exists a vertex with . Suppose that . Then, , and since , there exists an -coloring . For each , we can consider, without loss of generality, to be directed from to . Then, since and , . In other words, there is at least one color available to color . Thus, we can color all edges of . This contradiction completes the proof of theorem.

Data Availability

No data were used to support this study.

Disclosure

This research was performed as part of the employment of the author at Kharazmi University.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.