A Stability Theorem for Maximal $C_{2k+1}$-free Graphs

TL;DR

This paper proves that large maximal odd cycle-free graphs are structurally close to bipartite graphs, containing a large induced bipartite subgraph, and establishes the optimality of this result.

Contribution

It introduces a stability theorem for maximal $C_{2k+1}$-free graphs, showing their near bipartite structure when they have many edges, and proves this bound is tight.

Findings

Maximal $C_{2k+1}$-free graphs with many edges contain a large induced bipartite subgraph.

The structural property is optimal and cannot be improved.

The result generalizes and strengthens previous extremal graph theory results.

Abstract

For any positive integer $k$, we show that every maximal $C_{2k+1}$-free graph with at least $n^2/4-o(n^{3/2})$ edges contains an induced complete bipartite subgraph on $(1-o(1))n$ vertices. We also show that this is best possible.