The strong fractional choice number of series–parallel graphs
Introduction
A -fold colouring of a graph is a mapping which assigns to each vertex of a set of colours so that adjacent vertices receive disjoint colour sets. An -colouring of is a -fold colouring of such that for each vertex . The fractional chromatic number of is
An -list assignment of is a mapping which assigns to each vertex a set of permissible colours. A -fold-colouring of is a -fold colouring of such that for each vertex . We say is -choosable if for any -list assignment of , there is a -fold -colouring of . The choice number of is
The fractional choice number of is
It was proved by Alon, Tuza and Voigt [1] that for any finite graph , and moreover the infimum in the definition of is attained and hence can be replaced by the minimum. This implies that if is -colourable, then for some integer , is -choosable. The integer depends on and is usually a large integer. A natural question is for which , is -choosable for any positive integer . This motivated the definition of strong fractional choice number of a graph [6].
Definition 1 Assume is a graph and is a real number. We say is strongly fractional -choosable if for any positive integer , is -choosable. The strong fractional choice number of is The strong fractional choice number of a class of graphs is
It is known [5] that the infimum in the definition of is attained and hence can be replaced by the minimum. However, the supremum in the definition of maybe not attainable for some families of graphs.
It follows from the definition that for any graph , . The variant is intended to be a refinement for . However, currently we do not have a good upper bound for in terms of . It was conjectured by Erdős, Rubin and Taylor [3] that if is -choosable, then is -choosable for any positive integer . If this conjecture were true, then we would have . 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 is a-choosable graph. Is it true that for any positive integer , is-choosable?
If the answer of this question is affirmative, we also have . In any case, 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 denote the family of planar graphs and for a positive integer , let be the family of planar graphs containing no cycles of length . It was proved in [6] that and prove in [4] that .
The girth of a graph is the length of a shortest cycle in . If 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 , let This paper proves the following result.
Theorem 1 Assume is a positive integer, then .
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 by series parallel constructions.
Definition 2 A two-terminal series–parallel graph is defined recursively as follows: Let . Then is a two-terminal series–parallel graph. (The parallel construction) Let and 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.
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