The (2k-1)-connected multigraphs with at most k-1 disjoint cycles

In 1963, Corrádi and Hajnal proved that for all k≥1 and n≥3k, every (simple) graph G on n vertices with minimum degree δ(G)≥2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k− 1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the simple graphs G with minimum degree δ(G)≥ 2k−1 that do not contain k disjoint cycles. We use this result to answer Dirac’s question in full.