Coloring small locally sparse degenerate graphs and related problems

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.