A stability result for $C_{2k+1}$-free graphs

TL;DR

This paper extends a classical bipartiteness stability result for triangle-free graphs to $C_{2k+1}$-free graphs, showing near-bipartiteness under certain edge and vertex deletion conditions with optimal bounds.

Contribution

It generalizes a 1981 bipartiteness stability theorem from triangle-free graphs to odd cycle-free graphs, establishing tight bounds for near-bipartiteness.

Findings

Graphs with more than a specific number of edges are close to bipartite.

Near-bipartiteness can be achieved by deleting a limited number of vertices or edges.

The bounds for near-bipartiteness are proven to be optimal.

Abstract

A graph $G$ is called $C_{2k+1}$-free if it does not contain any cycle of length $2k+1$. In 1981, Haggkvist, Faudree and Schelp showed that every $n$-vertex triangle-free graph with more than $\frac{(n-1)^2}{4}+1$ edges is bipartite. In this paper, we extend their result and show that for $1\leq t\leq 2k-2$ and $n\geq 318t^2k$, every $n$-vertex $C_{2k+1}$-free graph with more than $\frac{(n-t-1)^2}{4}+\binom{t+2}{2}$ edges can be made bipartite by either deleting at most $t-1$ vertices or deleting at most $\binom{\lfloor\frac{t+2}{2}\rfloor}{2}+\binom{\lceil\frac{t+2}{2}\rceil}{2}-1$ edges. The construction shows that this is best possible.