Fractional colorings of partial $t$-trees with no large clique

Abstract

Dvo\v{r}\'ak and Kawarabayashi [European Journal of Combinatorics, 2017] 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 $2 \leq \omega \leq t$, then $\chi_f(G) \leq t + \frac{\omega - 1}{t}$, 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 $\chi_f(G) \geq t + 1 + \frac{1}{2}\log(1-2c) + o(1)$, which is approximately equal to the upper bound for small values $c$.