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

Abstract The Erdős–Gallai Theorem states that for k ≥ 3 , any n -vertex graph with no cycle of length at least k has at most 1 2 ( 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 , ⌊ k − 1 2 ⌋ ) } , where h ( n , k , a ) ≔ k − a 2 + 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 , ⌊ k − 1 2 ⌋ ) 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 , k − 3 2 ) } . The upper bound for e ( G ) here is tight.