A Stability Theorem for Maximal $K_{r+1}$-free Graphs

Kamil Popielarz; Julian Sahasrabudhe; Richard Snyder

A Stability Theorem for Maximal $K_{r+1}$-free Graphs

Combinatorics 2018-06-13 v3

Authors: Kamil Popielarz , Julian Sahasrabudhe , Richard Snyder

Abstract

For $r \geq 2$ , we show that every maximal $K_{r+1}$ -free graph $G$ on $n$ vertices with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contains a complete $r$ -partite subgraph on $(1 - o(1))n$ vertices. We also show that this is best possible. This result answers a question of Tyomkyn and Uzzell.

Keywords

hypergraph theory hadwiger conjecture bipartite graphs

Cite

@article{arxiv.1608.04675,
  title  = {A Stability Theorem for Maximal $K_{r+1}$-free Graphs},
  author = {Kamil Popielarz and Julian Sahasrabudhe and Richard Snyder},
  journal= {arXiv preprint arXiv:1608.04675},
  year   = {2018}
}

Comments

21 pages

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