On independent cycles and edges in graphs

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 K2. 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 ⩾ 3k − s. Justesen (1989) determined ex(n, k, 0) for all n ⩾ 3k and EX(n, k, 0) for all n > (13k − 4)/4, thereby settling a conjecture of Erdős and Pósa; further EX(n, k, k) was determined by Erdős and Gallai (n ⩾ 2k). 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 ⩾ 3k − s).