# A note on fractional covers of a graph

## DOI:

https://doi.org/10.26493/2590-9770.1398.fe9## Keywords:

fractional chromatic number, fractional cover, Kneser graph, n-colouring, a:b-colouring## Abstract

A fractional colouring of a graph *G* is a function that assigns a non-negative real value to all possible colour-classes of *G* containing any vertex of *G*, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of *G*. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of *G* belonging to a given class of graphs. The fractional chromatic number turns out to be a special instance of the fractional cover number. In this work we investigate fractional covers acting on (*k*+1)-clique-free subgraphs of *G* which, although sharing some similarities with fractional covers acting on *k*-colourable subgraphs of *G*, they exhibit some peculiarities. We first show that if a simple graph *G*_{2} is a homomorphic image of a simple graph *G*_{1}, then the fractional cover number defined on the (*k*+1)-clique-free subgraphs of *G*_{1} is bounded above by the corresponding number of *G*_{2}. We make use of this result to obtain bounds for the associated fractional cover number of graphs that are either *n*-colourable or *a* : *b*-colourable.