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 Kc−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(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:
- (i)
G is a subgraph of Wn,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 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. [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.