Counterexamples to Gerbner's Conjecture on Stability of Maximal $F$-free Graphs

TL;DR

This paper constructs counterexamples to Gerbner's conjecture, showing that certain maximal $F$-free graphs do not necessarily contain large induced complete $r$-partite subgraphs, thus disproving the conjecture for the case $r=2$.

Contribution

The authors provide explicit counterexamples to Gerbner's conjecture for $r=2$, demonstrating the conjecture does not hold in this case.

Findings

Counterexamples to Gerbner's conjecture for $r=2$

Maximal $F_{s,k}$-free graphs lack large induced complete bipartite subgraphs

Disproves the conjecture for the case $r=2$

Abstract

Let $F$ be an $(r+1)$-color critical graph with $r\geq 2$, that is, $\chi(F)=r+1$ and there is an edge $e$ in $F$ such that $\chi(F-e)=r$. Gerbner recently conjectured that every $n$-vertex maximal $F$-free graph with at least $(1-\frac{1}{r})\frac{n^2}{2}- o(n^{\frac{r+1}{r}})$ edges contains an induced complete $r$-partite graph on $n-o(n)$ vertices. Let $F_{s,k}$ be a graph obtained from $s$ copies of $C_{2k+1}$ by sharing a common edge. In this paper, we show that for all $k\geq 2$ if $G$ is an $n$-vertex maximal $F_{s,k}$-free graph with at least $n^{2}/4 - o(n^{\frac{s+2}{s+1}})$ edges, then $G$ contains an induced complete bipartite graph on $n-o(n)$ vertices. We also show that it is best possible. This disproves Gerbner's conjecture for $r=2$.