Tight bound on the minimum degree to guarantee graphs forbidding some odd cycles to be bipartite
Xiaoli Yuan, Yuejian Peng
TL;DR
The paper addresses the problem of determining the tight minimum-degree threshold for bipartiteness in graphs that forbid a family of odd cycles ${\mathcal{C}}$. It develops a stability-based framework around a shortest odd cycle and a detailed vertex-partitioning argument to show that for $2\le\ell<k$ and $n\ge 1000k^{8}$, if $G$ is ${\mathcal{C}}$-free and $\delta(G) > \max\{ \frac{n}{2(2\ell+1)}, \frac{2n}{2k+3} \}$, then $G$ is bipartite; at the threshold, the only non-bipartite ${\mathcal{C}}$-free graphs are the balanced blow-up $C_{2k+3}(n/(2k+3))$ or the blow-up $BC_{2\ell+1}(n)$. This unifies and extends the Andrásfai–Erdős–Sós, Häggkvist, and Yuan–Peng results and answers Erdős–Simonovits’s question for $r=2$ in the odd-cycle setting, providing a complete, tight characterization for large $n$ with explicit extremal constructions. The work thus advances a precise understanding of how minimum-degree conditions constrain bipartiteness under odd-cycle forbiddance and offers a unified perspective on stability phenomena in extremal graph theory.
Abstract
Erdős and Simonovits asked the following question: For an integer $r\geq 2$ and a family of non-bipartite graphs $\mathcal{H}$, determine the infimum of $α$ such that any $\mathcal{H}$-free $n$-vertex graph with minimum degree at least $αn$ has chromatic number at most $r$. We answer this question for $r=2$ and any family consisting of odd cycles. Let ${\mathcal C}$ be a family of odd cycles in which $C_{2\ell+1}$ is the shortest odd cycle not in ${\mathcal C}$ and $C_{2k+1}$ is the longest odd cycle in ${\mathcal C}$, we show that if $G$ is an $n$-vertex ${\mathcal C}$-free graph with $n\ge 1000k^{8}$ and $δ(G)>\max\{ n/(2(2\ell+1)), 2n/(2k+3)\}$, then $G$ is bipartite. Moreover, the bound of the minimum degree is tight.