Edge-disjoint cycles in regular directed graphs

Abstract

We prove that any k-regular directed graph with no parallel edges contains a collection of at least Ω(k) edge-disjoint cycles, conjecture that in fact any such graph contains a collection of at least ( k+1 2 ) disjoint cycles, and note that this holds for k ≤ 3. In this paper we consider the maximum size of a collection of edge-disjoint cycles in a directed graph. We pose the following conjecture: Conjecture 1: If G is a k-regular directed graph with no parallel edges, then G contains a collection of ( k+1 2 ) edge-disjoint cycles. We prove two weaker results: Theorem 1: If G is a k-regular directed graph with no parallel edges, then G contains a collection of at least 5k/2− 2 edge-disjoint cycles. Theorem 2: If G is a k-regular directed graph with no parallel edges, then G contains a collection of at least k edge-disjoint cycles, where = 3 219 . ∗AT & T Bell Labs, Murray Hill, NJ 07974, USA and Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Research supported in part by a United States Israel BSF Grant. †Faculty of Mathematical Sciences, University of Oxford. ‡Department of Mathematics, Carnegie Mellon University and Department of Computer Science, University of Toronto. Research supported in part by an NSERC fellowship.