On independent cycles and edges in graphs

Abstract

For integers k, s with 0 ⩽ s ⩽ k, let G(n, k, s) be the class of graphs on n vertices not containing k independent (i.e., vertex disjoint) subgraphs of which k − s are cycles and the remaining are complete graphs K 2. Let EX( n, k, s) be the set of members of G(n, k, s) with the maximum number of edges and denote the number of edges of a graph in EX( n, k, s) by ex( n, k, s); to avoid trivialities, assume k ⩾ 2 and n ⩾ 3 k − s. Justesen (1989) determined ex( n, k, 0) for all n ⩾ 3 k and EX( n, k, 0) for all n > (13 k − 4)/4, thereby settling a conjecture of Erdős and Pósa; further EX( n, k, k) was determined by Erdős and Gallai ( n ⩾ 2 k). In the present paper, by modifying the argument presented by Justesen, we determine EX( n, k, s) for all n, k, s (0 ⩽ s ⩽ k, k ⩾ 2, n ⩾ 3 k − s).