no code implementations • 22 Feb 2021 • Freddie Illingworth
Given an $(r + 1)$-chromatic graph $H$, the fundamental edge stability result of Erd\H{o}s and Simonovits says that all $n$-vertex $H$-free graphs have at most $(1 - 1/r + o(1)) \binom{n}{2}$ edges, and any $H$-free graph with that many edges can be made $r$-partite by deleting $o(n^{2})$ edges.
Combinatorics 05C15, 05C35
no code implementations • 18 Dec 2020 • Freddie Illingworth
Locally bipartite graphs, first mentioned by Luczak and Thomass\'{e}, are the natural variant of triangle-free graphs in which each neighbourhood is bipartite.
Combinatorics 05C15, 05C35
