Coloring small locally sparse degenerate graphs and related problems

TL;DR

This paper investigates the chromatic number of small, locally sparse, degenerate graphs, establishing bounds on their size and coloring properties, and explores implications for Hadwiger's conjecture and online coloring.

Contribution

It provides new bounds on the minimum order of triangle-free degenerate graphs with high chromatic number and links these bounds to online chromatic numbers, advancing understanding of graph coloring complexities.

Findings

Lower bound on minimum order of such graphs grows exponentially with d.

New upper bound on the chromatic number of triangle-free d-degenerate graphs.

Improved asymptotic bounds for the online chromatic number g(n).

Abstract

The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and Zykov from the early 50s show that there exist $d$-degenerate $(d+1)$-chromatic graphs that are triangle-free, however these constructions grow rapidly with $d$. Motivated by this and addressing a problem posed by the second author at the Oberwolfach Graph Theory workshop, we prove that the minimum order $f(d)$ of a $d$-degenerate triangle-free graph of chromatic number $d+1$ satisfies $e^{\Omega(d)}\le f(d)\le e^{O(d^2\log d)}.$ The lower bound follows from a novel upper bound on the chromatic number of triangle-free graphs: Every triangle-free $d$-degenerate graph $G$ on $n \le e^{O(d)}$ vertices satisfies $$\chi(G)\le O\left(\frac{d}{\log\left(d/\log n\right)}\right).$$ We extend this to a more general result about degenerate graphs with sparse neighborhoods, which has applications to many graph coloring problems: For example, we prove that every counterexample to Hadwiger's conjecture with parameter $t$ must have a complete bipartite subgraph with one exponentially large side ($K_{a,b}$ where $a=(\log t)^{1/2-o(1)}$ and $b=e^{t^{1-o(1)}}$) or a small and very dense subgraph (of order $\le t$ with $t^{2-o(1)}$ edges) in some neighborhood. For the upper bound on $f(d)$ we establish a surprising connection between $f(d)$ and the on-line-chromatic number $g(n)$ of $n$-vertex triangle-free graphs. We also give an asymptotic improvement of the previous best upper bound for $g(n)$ due to Lov\'{a}sz, Saks and Trotter from 1989. Along the way we disprove a generalization of Harris' fractional coloring conjecture to graphs of bounded clique number and raise numerous problems which open up interesting directions to explore for future research.