A Strong Structural Stability of C 2 k + 1 -Free Graphs

Abstract

Füredi and Gunderson showed that $ex\left(n,C_{2k+1}\right)$ 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. If a graph $G$ and a graph $H$ have at most one vertex in common and there is no edge connecting $V\left(G\right)-V\left(G\right)\cap V\left(H\right)$ and $V\left(H\right)-V\left(G\right)\cap V\left(H\right)$, then we call graph $H$ a suspension to graph $G$ with $\mid V\left(G\right)\cap V\left(H\right)\mid$ suspension point. Let $T^{*}\left(r,n\right)$ be obtained by adding a suspension $K_r$ with 1 suspension point to $K_{\lfloor \frac{n-r+1}{2}\rfloor ,\lceil \frac{n-r+1}{2}\rceil }$. Let $d_2\left(G\right)=\min \left\{\mid T\mid :T\subseteq V\left(G\right),G-T\text{is bipartite}\right\}$ and ${\gamma }_2\left(G\right)=\min \left\{\mid E\mid :E\subseteq E\left(G\right),G-E\text{is bipartite}\right\}$. Ren, Wang, Wang, and Yang showed that if $G$ is an $n$-vertex $C_{2k+1}$-free graph with $e\left(G\right)\ge \left(\frac{{\left(n-r+1\right)}^2}{4}\right)+\left(\frac{r}{2}\right)$, then $d_2\left(G\right)\le r-2$ and ${\gamma }_2\left(G\right)\le \left(\frac{\lceil \frac{r}{2}\rceil }{2}\right)+\left(\frac{\lfloor \frac{r}{2}\rfloor }{2}\right)$, and equalities hold if and only if $G=T^{*}\left(r,n\right)$ for $3\le r\le 2k$ and $n\ge 318{\left(r-2\right)}^{2}k$. In this paper, we show that for integers $r,k$ with $3\le r\le 2k-4$ and $n\ge 20{\left(r+2\right)}^{2}k$, if $G$ is a $C_{2k+1}$-free $n$-vertex graph with $e\left(G\right)\ge e\left(T^{*}\left(r,n\right)\right)$, then $G$ is obtained by adding suspensions to a ‘nearly balanced complete’ bipartite graph $G\left[V_1,V_2\right]$ one by one and the number of vertices not in $G\left[V_1,V_2\right]$ is no more than $r-1$. Furthermore, the total number of vertices not in $G\left[V_1,V_2\right]$ equals $r-1$ if and only if $G=T^{*}\left(r,n\right)$. Roughly speaking, we give a strong structural information for $C_{2k+1}$-free graph $G$ rather than the distance from being bipartite when $ex\left(n,C_{2k+1}\right)-e\left(G\right)=O\left(nk\right)$ if $k=O\left(n^{\frac{1}{3}}\right)$. In the proof, 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.