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

TL;DR

This paper characterizes (2k-1)-connected multigraphs that contain at most k-1 disjoint cycles, extending classical results on cycle packings and connectivity in graphs.

Contribution

It provides a complete characterization of (2k-1)-connected multigraphs lacking k disjoint cycles, building on recent results for simple graphs.

Findings

Characterization of (2k-1)-connected multigraphs with fewer than k disjoint cycles

Extension of classical cycle packing results to multigraphs

Resolution of Dirac's question on multigraphs without k disjoint cycles

Abstract

In 1963, Corr\'adi and Hajnal proved that for all $k \ge 1$ and $n \ge 3k$, every (simple) graph on n vertices with minimum degree at least 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 $\delta(G) \ge 2k-1$ that do not contain k disjoint cycles. We use this result to answer Dirac's question in full.