Mon DI-fusion | À propos de DI-fusion | Contact |  # Bipartite Induced Density in Triangle-Free Graphs

Article révisé par les pairs
 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\$.