Fractional colorings of partial t-trees with no large clique

Abstract

Dvořák and Kawarabayashi [2] asked, what is the largest chromatic number attainable by a graph of treewidth t with no $K_r$ subgraph? In this paper, we consider the fractional version of this question. We prove that if G has treewidth t and clique number $2\le \omega \le t$, then ${\chi }_f\left(G\right)\le t+\frac{\omega -1}{t}$, and we show that this bound is tight for $\omega =t$. We also show that for each value $0<c<\frac{1}{2}$, there exists a graph G of a large treewidth t and clique number $\omega =\lfloor (1-c)t\rfloor$ satisfying ${\chi }_f\left(G\right)\ge t+1+\frac{1}{2}\log (1-2c)+o\left(1\right)$, which is approximately equal to the upper bound for small values c.