Stability in the Erdős–Gallai Theorem on cycles and paths, II

The Erdős–Gallai Theorem states that for k≥3, any n-vertex graph with no cycle of length at least k has at most [Formula preseted](k−1)(n−1) edges. A stronger version of the Erdős–Gallai Theorem was given by Kopylov: If G is a 2-connected n-vertex graph with no cycle of length at least k, then e(G)≤max{h(n,k,2),h(n,k,⌊[Formula preseted]⌋)}, where h(n,k,a)≔[Formula preseted]+a(n−k+a). Furthermore, Kopylov presented the two possible extremal graphs, one with h(n,k,2) edges and one with h(n,k,⌊[Formula preseted]⌋) edges. In this paper, we complete a stability theorem which strengthens Kopylov's result. In particular, we show that for k≥3 odd and all n≥k, every n-vertex 2-connected graph G with no cycle of length at least k is a subgraph of one of the two extremal graphs or e(G)≤max{h(n,k,3),h(n,k,[Formula preseted])}. The upper bound for e(G) here is tight.