A stronger result on fractional strong colourings

TL;DR

This paper extends a recent fractional strong coloring result by providing a generalized probabilistic method applicable to graphs with specific vertex partition properties, potentially aiding in bounding chromatic numbers.

Contribution

It generalizes the fractional strong coloring theorem to broader graph classes with vertex partitions and neighbor constraints.

Findings

Established a probability distribution on stable sets hitting each vertex set with specified probability.

Generalized fractional strong coloring results to graphs with partitioned vertices and neighbor restrictions.

Potential applications in probabilistic bounds for chromatic and fractional chromatic numbers.

Abstract

Aharoni, Berger and Ziv recently proved the fractional relaxation of the strong colouring conjecture. In this note we generalize their result as follows. Let $k\geq 1$ and partition the vertices of a graph $G$ into sets $V_1,..., V_r$, such that for $1\leq i \leq r$ every vertex in $V_i$ has at most $\max\{k, |V_i|-k \}$ neighbours outside $V_i$. Then there is a probability distribution on the stable sets of $G$ such that a stable set drawn from this distribution hits each vertex in $V_i$ with probability $1/|V_i|$, for $1\leq i\leq r$. We believe that this result will be useful as a tool in probabilistic approaches to bounding the chromatic number and fractional chromatic number.