Bipartite induced density in triangle-free graphs

TL;DR

This paper investigates properties of triangle-free graphs, establishing bounds on bipartite induced subgraphs, fractional chromatic number, and list chromatic number, with results that are sharp up to logarithmic or constant factors.

Contribution

It proves new bounds on bipartite induced subgraphs and chromatic numbers in triangle-free graphs, and proposes conjectures for further tightening these bounds.

Findings

Bipartite induced subgraph with minimum degree at least $d^2/(2n)$ exists in such graphs.

Fractional chromatic number is at most $ ext{min}igrace{n/d, (2+o(1)) ext{sqrt}(n/ ext{log} n)igrace$.

List chromatic number is at most $O( ext{min}igrace{ ext{sqrt}(n), (n ext{log} n)/digrace$.

Abstract

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.