# Stability Results on the Circumference of a Graph

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## 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 $$\max \left\{{f\left({n,2,c} \right),\,f\left({n,\,\left\lfloor {{c \over 2}} \right\rfloor ,\;c} \right)} \right\}$$ edges, with equality if and only if G is isomorphic to Wn,2,c or $${W_{_{n,\left\lfloor {{c \over 2}} \right\rfloor ,\,c}}}$$. 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 $$e\left(G \right) > \max \left\{{f\left({n,\,3,\,c} \right),\;f\left({n,\;\left\lfloor {{c \over 2}} \right\rfloor - 1,\;c} \right)} \right\}$$, then either G is a subgraph of Wn,2,c or $${W_{n,\left\lfloor {{c \over 2}} \right\rfloor ,c}}$$, or c is odd and G is a subgraph of a member of two well-characterized families which we define as χn,c and γn,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 $${W_{n,\left\lfloor {{c \over 2}} \right\rfloor ,c}}$$, then one of the following holds:

1. (i)

G is a subgraph of Wn, k, c or $${{\cal X}_{n,c}} \cup {{\cal Y}_{n,c}}$$

2. (ii)

k = 2, c is odd, and G is a subgraph of a member of χn,c∪γn,c, or

3. (iii)

k ≥ 3 and G is a subgraph of the union of a clique Kc−k+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. 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.

## Author information

Correspondence to Jie Ma or Bo Ning.