A strong structural stability of $C_{2k+1}$-free graphs

TL;DR

This paper establishes a structural stability theorem for $C_{2k+1}$-free graphs, showing they are close to bipartite graphs with controlled suspensions, and introduces the concept of strong-$2k$-core for simplified proofs.

Contribution

It provides a new structural stability result for odd cycle-free graphs, introducing the strong-$2k$-core concept for simpler proofs and tighter bounds.

Findings

Graphs close to extremal are constructed by suspensions on bipartite graphs.

The number of suspension vertices is bounded by r-1.

The results improve bounds on vertex and edge modifications needed for bipartiteness.

Abstract

F\"uredi and Gunderson showed that $ex(n, C_{2k+1})$ is achieved only on $K_{\lfloor\frac{n}{2}\rfloor, \lceil\frac{n}{2}\rceil}$ if $n\ge 4k-2$. It is natural to study how far a $ C_{2k+1}$-free graph is from being bipartite.Let $T^*(r, n)$ be obtained by adding a suspension $K_{r}$ with $1$ suspension point to $K_{\lfloor\frac{n-r+1}{2}\rfloor, \lceil\frac{n-r+1}{2}\rceil}$. We show that for integers $r, k$ with $3\le r\le 2k-4$ and $n\ge 20(r+2)^2k$, if $G$ is a $C_{2k+1}$-free $n$-vertex graph with $e(G)\ge e(T^*(r, n))$, then $G$ is obtained by adding suspensions to a bipartite graph one by one and the total number of vertices in all suspensions minus intersection points is no more than $r-1$. In other words, $G=B\bigcup\limits_{i=1}^p G_i$, where $B$ is a bipartite graph, $G_1$ is a suspension to $B$, $G_j$ is a suspension to $B\bigcup\limits_{i=1}^{j-1} G_i$ for $2\le j\le p$ and $\sum\limits_{i=1}^p \vert V(G_i)-V(G_i)\cap V(B\bigcup\limits_{i=1}^{j-1} G_i) \vert\le r-1$. Furthermore, $\sum\limits_{i=1}^p \vert V(G_i)-V(G_i)\cap V(B\bigcup\limits_{i=1}^{j-1} G_i) \vert= r-1$ if and only if $G=T^*(r, n)$. Let $d_2(G)=\min\{|T|: T\subseteq V(G), G-T \ \text{is bipartite}\}$ and $\gamma_2(G)=\min\{|E|: E\subseteq E(G), G-E \ \text{is bipartite}\}$. Our structural stability result implies that $d_2(G)\le r-1$ and $\gamma_2(G)\le {\lceil\frac{r}{2}\rceil \choose 2}+{\lfloor\frac{r}{2}\rfloor \choose 2}$ under the same condition, which is a recent result of Ren-Wang-Wang-Yang [SIAM J. Discrete Math. 38 (2024)]. They proved $d_2(G)\le r-1$ and $\gamma_2(G)\le {\lceil\frac{r}{2}\rceil \choose 2}+{\lfloor\frac{r}{2}\rfloor \choose 2}$ separately. We introduce a new concept strong-$2k$-core which is the key that we can give a stronger structural stability result but a simpler proof.