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

TL;DR

This paper investigates the fractional chromatic number of graphs with bounded treewidth and clique number, establishing tight bounds and constructing graphs that nearly attain these bounds, advancing understanding of graph coloring constraints.

Contribution

It provides tight bounds on fractional chromatic numbers for graphs with given treewidth and clique number, and constructs graphs that nearly reach these bounds, extending prior combinatorial results.

Findings

Bound $oxed{ ext{for } ext{treewidth } t ext{ and clique number } oldsymbol{ ext{omega}} ext{, } oldsymbol{ ext{chi}_f(G)} oldsymbol{ ext{ } oldsymbol{ ext{leq}} oldsymbol{t + rac{ ext{omega} - 1}{t}}}$.

Existence of graphs with large treewidth and clique number approaching $t$, with fractional chromatic number close to the upper bound, demonstrating tightness.

Approximate characterization of fractional chromatic number for graphs with clique number proportional to treewidth, for small proportional constants.

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 $c$.