Stability results on the circumference of a graph

TL;DR

This paper extends stability results on graphs with a given circumference, generalizing previous theorems and characterizing the structure of 2-connected graphs with many edges relative to their circumference.

Contribution

It provides a unified generalization of existing stability theorems on graphs with specified circumference, including new structural characterizations.

Findings

Characterization of extremal graphs with maximum edges for given circumference.

Generalization of F"uredi et al.'s stability theorem to broader classes of graphs.

Proof of a stability result related to Bondy's classical theorem.

Abstract

In this paper, we extend and refine previous Tur\'an-type results on graphs with a given circumference. Let $W_{n,k,c}$ be the graph obtained from a clique $K_{c-k+1}$ by adding $n-(c-k+1)$ isolated vertices each joined to the same $k$ vertices of the clique, and let $f(n,k,c)=e(W_{n,k,c})$. Improving a celebrated theorem of Erd\H{o}s and Gallai, Kopylov proved that for $c<n$, any 2-connected graph $G$ on $n$ vertices with circumference $c$ has at most $\max{f(n,2,c),f(n,\lfloor\frac{c}{2}\rfloor,c)}$ edges. Recently, F\"uredi et al. proved a stability version of Kopylov's theorem. Their main result states that if $G$ is a 2-connected graph on $n$ vertices with circumference $c$ such that $10\leq c<n$ and $e(G)>\max{f(n,3,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)}$, then either $G$ is a subgraph of $W_{n,2,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, or $c$ is odd and $G$ is a subgraph of a member of two well-characterized families which we define as $\mathcal{X}_{n,c}$ and $\mathcal{Y}_{n,c}$. We prove that if $G$ is a 2-connected graph on $n$ vertices with minimum degree at least $k$ and circumference $c$ such that $10\leq c<n$ and $e(G)>\max{f(n,k+1,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)}$, then one of the following holds: (i) $G$ is a subgraph of $W_{n,k,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, (ii) $k=2$, $c$ is odd, and $G$ is a subgraph of a member of $\mathcal{X}_{n,c}\cup \mathcal{Y}_{n,c}$, or (iii) $k\geq 3$ and $G$ is a subgraph of the union of a clique $K_{c-k+1}$ and some cliques $K_{k+1}$'s, where any two cliques share the same two vertices. This provides a unified generalization of the above result of F\"uredi et al. as well as a recent result of Li et al. and independently, of F\"uredi et al. on non-Hamiltonian graphs. Moreover, we prove a stability result on a classical theorem of Bondy on the circumference.