# Stability Results on the Circumference of a Graph

## Abstract

In this paper, we extend and refine previous Turán-type results on graphs with a given circumference. Let Wn,k,c be the graph obtained from a clique Kck+1 by adding n − (ck + 1) isolated vertices each joined to the same k vertices of the clique, and let f(n,k,c) = e(Wn,k,c). Improving a celebrated theorem of Erdős and Gallai , Kopylov  proved that for c<n, any 2-connected graph G on n vertices with circumference c has at most $$\max \left\{f(n, 2, c), f\left(n,\left\lfloor\frac{c}{2}\right\rfloor, c\right)\right\}$$ edges, with equality if and only if G is isomorphic to Wn,2,c or $${W_{n,2,c}}$$. Recently, Füredi et al. [15,14] 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 ≤ c<n and $$W_{n,\left\lfloor\frac{c}{2}\right\rfloor, c}$$, then either G is a subgraph of Wn,2,c or $$e\left( G \right) > \max \left\{ {f(n,3,c),f\left( {n,\left\lfloor {{c \over 2},} \right\rfloor - 1,c} \right)} \right\}$$, or c is odd and G is a subgraph of a member of two well-characterized families which we define as $$e(G)>\max \left\{f(n, k+1, c), f\left(n,\left\lfloor\frac{c}{2}\right\rfloor-1, c\right)\right\}$$ and $$W_{n,\left\lfloor\frac{c}{2}\right\rfloor, c}$$.

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 ≤ c<n and $$\mathcal{X}_{n, c} \cup \mathcal{Y}_{n, c}$$, then one of the following holds:

1. (i)

G is a subgraph of Wn,k,c or $${W_{n,2,c}}$$

2. (ii)

k = 2, c is odd, and G is a subgraph of a member of $$W_{n,\left\lfloor\frac{c}{2}\right\rfloor, c}$$, or

3. (iii)

k ≤ 3 and G is a subgraph of the union of a clique Kck+1 and some cliques Kk+1’s, where any two cliques share the same two vertices.

This provides a unified generalization of the above result of Füredi et al. [15,14] as well as a recent result of Li et al.  and independently, of Füredi et al.  on non-Hamiltonian graphs. graphs. A refinement and some variants of this result are also obtained. Moreover, we prove a stability result on a classical theorem of Bondy  on the circumference. We use a novel approach, which combines several proof ideas including a closure operation and an edge-switching technique. We will also discuss some potential applications of this approach for future research.

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## Acknowledgement

The first author would like to thank Alexandr V. Kostochka for helpful discussions.

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Correspondence to Jie Ma or Bo Ning.