Planar graphs without cycles of length 4 or 5 are -colorable
Introduction
A function that assigns sets to all vertices of a graph is a set coloring if the sets assigned to adjacent vertices are disjoint. For positive integers and , an -coloring of a graph is a set coloring with range , i.e., a set coloring that to each vertex assigns a -element subset of . The concept of -coloring is a generalization of the conventional vertex coloring. In fact, an -coloring is exactly an ordinary proper -coloring.
The fractional chromatic number of , denoted by , is the infimum of the fractions such that admits an -coloring. Note that for any graph , where is the chromatic number of . The fractional coloring was first introduced in 1973 [10] to seek for a proof of the Four Color Problem. Since then, it has been the focus of many intensive research efforts, see [12]. In particular, fractional coloring of planar graphs without cycles of certain lengths is widely studied. Pirnazar and Ullman [11] showed that the fractional chromatic number of a planar graph with girth at least is at most . Dvořák et al. [7] showed that every planar graph of odd-girth at least 9 is -colorable. Recently, Dvořák et al. [6] showed that every planar triangle-free graph on vertices is -colorable, and thus it has fractional chromatic number at most .
Well-known Steinberg’s Conjecture asserts that every planar graph without cycles of length 4 or 5 is 3-colorable. Recently, Steinberg’s conjecture was disproved [4]. This conjecture, though disproved, had motivated a lot of research, see [3]. Since for any graph , it is natural to ask whether there exists a constant such that for all planar graphs without cycles of length 4 or 5. In this paper, we confirm this is the case for . In fact, we prove the following stronger theorem.
Theorem 1.1 Every planar graph without cycles of length 4 or 5 is-colorable, and thus its fractional chromatic number is at most .
The independence number of a graph is the size of a largest independent set in . The independence ratio of is the quantity . The famous Four Color Theorem [2] implies that every planar graph has independence ratio at least . In 1976, Albertson [1] proved a weaker result that every planar graph has independence ratio at least without using the Four Color Theorem. In 2016, Cranston and Rabern [5] improved this constant to . If is a triangle-free planar graph, a classical theorem of Grőtzsch [9] says that is 3-colorable, and thus has independence ratio at least . This bound can be slightly improved—Steinberg and Tovey [13] proved that the independence ratio is at least , and gave an infinite family of planar triangle-free graphs for that this bound is tight. Steinberg’s Conjecture would imply that every planar graph without cycles of length 4 or 5 has independence ratio at least , and it is not known whether this weaker statement holds or not. Since for any graph , we have the following corollary by Theorem 1.1.
Corollary 1.2 Every planar graph without cycles of length 4 or 5 has independence ratio at least.
It is not clear whether the constant from Theorem 1.1 is the best possible, and we suspect this is not the case. Hence, the following question is of interest.
Problem 1.3 What is the infimum of fractional chromatic numbers of planar graphs without cycles of length 4 or 5?
Let us remark that the counterexample to Steinberg’s conjecture constructed in [4] is -colorable, and thus we cannot even exclude the possibility that the answer is .
The proof of Theorem 1.1 naturally proceeds in list coloring setting. A list assignment for a graph is a function that to each vertex of assigns a set of colors. A set coloring of is an -set coloring if for all . For a positive integer , we say that is an -coloring of if is an -set coloring and for all . If such an -coloring exists, we say that is -colorable. For an integer , we say that is -choosable if is -colorable from any assignment of lists of size . We actually prove the following strengthening of Theorem 1.1.
Theorem 1.4 Every planar graph without cycles of length 4 or 5 is-choosable.
Section snippets
Colorability of small graphs
Let us start with some technical results on list-colorability of small graphs, especially paths and cycles. In the proofs, it is convenient to work with a non-uniform version of set coloring. Let be an arbitrary function. An -coloring of a graph is an -set coloring such that for all . If such an -coloring exists, we say that is -colorable. We repeatedly use the following simple observation.
Lemma 2.1 Let be an assignment of lists to vertices of a graph
Properties of a minimal counterexample
We are going to prove a mild strengthening of Theorem 1.4 where a clique (one vertex, two adjacent vertices, or a triangle) is precolored. A (hypothetical) counterexample (to this strengthening) is a triple , where is a plane graph without - or -cycles, is the vertex set of a clique of , and is an assignment of lists of size to vertices of and pairwise disjoint lists of size to vertices , such that is not -colorable. The order of the counterexample is the
Notation
Consider a minimal counterexample . We say that the faces of of length at least are -faces. Since is -connected by Lemma 3.1, every face of is bounded by a cycle, and in particular, every face of is either a -face or a -face. A vertex is a -vertex if is internal and . We say that is a -vertex if either or .
Let be a part of the cycle bounding a -face of , and for , let be the face incident with the edge . If both
-colorability of planar graphs
We are now ready to prove our main result.
Proof of Theorem 1.4 Suppose for a contradiction that there exists a plane graph without - or -cycles and an assignment of lists of size 11 to vertices of such that is not -colorable. Let be any vertex of , let be any -element subset of , and let for all . Then is not -colorable, and thus is a counterexample. Therefore, there exists a minimal counterexample . Let be the assignment of
Acknowledgments
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement n.616787. Xiaolan Hu is partially supported by NSFC, China under Grant Number 11601176 and NSF of Hubei Province, China under Grant Number 2016CFB146.
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