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 [8], Kopylov [18] 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. [20] and independently, of Füredi et al. [12] 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 [2] 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.

Research supported in part by National Natural Science Foundation of China grants 11501539 and 11622110 and Anhui Initiative in Quantum Information Technologies grant AHY150200.

Research supported in part by National Natural Science Foundation of China grant 11601379.

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Ma, J., Ning, B. Stability Results on the Circumference of a Graph. Combinatorica 40, 105–147 (2020). https://doi.org/10.1007/s00493-019-3843-4