Critical graphs without triangles: an optimum density construction

TL;DR

This paper constructs dense, triangle-free, chromatic-critical graphs with optimal edge density for all chromatic numbers $k \\geq 4$, and extends to graphs avoiding small odd cycles, establishing their maximal density.

Contribution

It provides explicit constructions of dense, triangle-free, chromatic-critical graphs for all $k \\geq 4$, and demonstrates the existence of dense graphs avoiding small odd cycles with optimal density.

Findings

Constructed dense, triangle-free, $k$-critical graphs for all $k \\geq 4$.

Achieved asymptotically optimal edge density for $k \\geq 6$.

Extended results to graphs avoiding all small odd cycles with maximal density.

Abstract

We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\geq 4$. For $k\geq 6$ our constructions have $> (\frac{1}{4} -\varepsilon)n^2$ edges, which is asymptotically best possible by Tur\'an's theorem. We also demonstrate (nonconstructively) the existence of dense $k$-critical graphs avoiding all odd cycles of length $\leq \ell$ for any $\ell$ and any $k\geq 4$, again with a best possible density of $>(\frac{1}{4} -\varepsilon)n^2$ edges for $k\geq 6$. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the correct maximal density of $k$-critical members ($k\geq 6$).