The strong fractional choice number of series–parallel graphs

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Abstract

The strong fractional choice number of a graph G is the infimum of those real numbers r such that G is (rm,m)-choosable for every positive integer m. The strong fractional choice number of a family G of graphs is the supremum of the strong fractional choice number of graphs in G. We denote by Qk the class of series–parallel graphs with girth at least k. This paper proves that the strong fractional choice number of Qk equals 2+1(k+1)4.

Introduction

A b-fold colouring of a graph G is a mapping ϕ which assigns to each vertex v of G a set ϕ(v) of b colours so that adjacent vertices receive disjoint colour sets. An (a,b)-colouring of G is a b-fold colouring ϕ of G such that ϕ(v){1,2,,a} for each vertex v. The fractional chromatic number of G is χf(G)=inf{ab:G is (a,b)-colourable}.

An a-list assignment of G is a mapping L which assigns to each vertex v a set L(v) of a permissible colours. A b-foldL-colouring of G is a b-fold colouring ϕ of G such that ϕ(v)L(v) for each vertex v. We say G is (a,b)-choosable if for any a-list assignment L of G, there is a b-fold L-colouring of G. The choice number of G is ch(G)=min{a:G is (a,1)-choosable}

The fractional choice number of G is chf(G)=inf{ab:G is (a,b)-choosable}.

It was proved by Alon, Tuza and Voigt [1] that for any finite graph G, χf(G)=chf(G) and moreover the infimum in the definition of chf(G) is attained and hence can be replaced by the minimum. This implies that if G is (a,b)-colourable, then for some integer m, G is (am,bm)-choosable. The integer m depends on G and is usually a large integer. A natural question is for which (a,b), G is (am,bm)-choosable for any positive integer m. This motivated the definition of strong fractional choice number of a graph [6].

Definition 1

Assume G is a graph and r is a real number. We say G is strongly fractional r-choosable if for any positive integer m, G is (rm,m)-choosable. The strong fractional choice number of G is chf(G)=inf{r:G is strongly fractional r-choosable}.The strong fractional choice number of a class G of graphs is chf(G)=sup{chf(G):GG}.

It is known [5] that the infimum in the definition of chf(G) is attained and hence can be replaced by the minimum. However, the supremum in the definition of chf(G) maybe not attainable for some families G of graphs.

It follows from the definition that for any graph G, chf(G)ch(G)1. The variant chf(G) is intended to be a refinement for ch(G). However, currently we do not have a good upper bound for chf(G) in terms of ch(G). It was conjectured by Erdős, Rubin and Taylor [3] that if G is k-choosable, then G is (km,m)-choosable for any positive integer m. If this conjecture were true, then we would have chf(G)ch(G). But this conjecture is refuted recently by Dvořák, Hu and Sereni in [2]. Maybe the conjecture of Erdős, Rubin and Taylor is not too far from the truth. As a modification of this conjecture, we put forward the following question.

Question 1

Assume G is ak-choosable graph. Is it true that for any positive integer m,G is(km+1,m)-choosable?

If the answer of this question is affirmative, we also have chf(G)ch(G). In any case, chf(G) is an interesting graph invariant and there are many challenging open problems concerning this parameter. The strong fractional choice number of planar graphs were studied in [6] and [4]. Let P denote the family of planar graphs and for a positive integer k, let Pk be the family of planar graphs containing no cycles of length k. It was proved in [6] that 5chf(P)4+29 and prove in [4] that 4chf(P3)3+117.

The girth of a graph G is the length of a shortest cycle in G. If G has no cycle, then its girth is infinite. In this paper, we consider the strong fractional choice number of series–parallel graphs with given girth. For a positive integer k, let Qk={G:G is a series–parallel graph with girth at least k}.This paper proves the following result.

Theorem 1

Assume k is a positive integer, then chf(Qk)=2+1(k+1)4.

Section snippets

The proof of Theorem 1

The family of series–parallel graphs is a well studied family of graphs and there are many equivalent definitions for this class of graphs. For the purpose of using induction, we adopt the definition that recursively construct series–parallel graphs from K2 by series parallel constructions.

Definition 2

A two-terminal series–parallel graph (G;x,y) is defined recursively as follows:

  • Let V(K2)={0,1}. Then (K2;0,1) is a two-terminal series–parallel graph.

  • (The parallel construction) Let (G;x,y) and (G;x,y) be

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (6)

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1

Grant Numbers: NSFC, China 11971438, NSFC, China 11771403 and 111 project of Ministry of Education of China .

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