Choosability and fractional chromatic numbers

Abstract

A graph G is (a, b)-choosable if for any assignment of a list of a colors to each of its vertices there is a subset of b colors of each list so that subsets corresponding to adjacent vertices are disjoint. It is shown that for every graph G, the minimum ratio a/b where a, b range over all pairs of integers for which G is (a, b)-choosable is equal to the fractional chromatic number of G.

Graph Imperfection

“…In Section 4 we exhibit examples such that certain coordinates have to be large for any vector x attaining the maximum in (1), that is, for any nonnegative integer-valued vector x with imp(…”

List colorings with measurable sets

“…As mentioned in Introduction Section, Alon et al [1] showed that the fractional chromatic number and the list fractional chromatic number are the same. Here, we follow the lines of their proof and show that the fractional chromatic and the measurable list chromatic numbers are the same.…”

“…A list variant of fractional colorings was studied by Alon et al [1]. Fix a graph G. For each positive integer q, let p q be the smallest integer such that if every vertex v is assigned a list L(v) of p q colors, then there exists a q-coloring c of G with c(v) ⊆ L(v).…”

“…The list fractional chromatic number χ f, (G) of a graph G is the limit of the ratio p q /q when q tends to infinity. Alon et al [1] showed that if G is finite then χ f, (G) exists and is equal to χ f (G). It seems unavoidable to consider the limit when q tends to infinity in this definition.…”

Extending Fractional Precolorings

“…Since the fractional chromatic number and the fractional list chromatic number of a graph are always equal [8] (also see [12,Theorem 3.8.1]), it follows that any such precoloring can be extended for ε ≥ 1 if d ≥ 3; thus, the minimum value is always at most one.…”