Stability Results on the Circumference of a Graph

In this paper, we extend and refine previous Turán-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 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 W n ,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 W n ,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: (i) G is a subgraph of W n,k,c or $${W_{n,2,c}}$$ (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 (iii) k ≤ 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 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.