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Bipartite Induced Density in Triangle-Free Graphs

par Cames Van Batenburg, Wouter ;De Joannis de Verclos, Rémi;Kang, Ross J.;Pirot, François
Référence The electronic journal of combinatorics, 27, 2
Publication Publié, 2020-04

Article révisé par les pairs

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Résumé :

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 $n oinfty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(min{sqrt{n},(nlog n)/d})$ as $n oinfty$.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 $n oinfty$. Second, any  triangle-free graph on $n$ vertices has list chromatic number at most $O(sqrt{n/log n})$ as $n oinfty$.