Integer 4-Flows and Cycle Covers

Abstract

Let G be a bridgeless graph and denote by cc(G) the shortest length of a cycle cover of G. Let V2(G) be the set of vertices of degree 2 in G. It is known that if cc(G)≤1.4|E(G)| for every bridgeless graph G with |V2(G)|≤ $$\frac{1}{10}$$ |E(G)|, then the Cycle Double Cover Conjecture is true. The best known result cc(G)≤ $$\frac{5}{3}$$ |E(G)| (≈1.6667|E(G)|) was established over 30 years ago. Recently, it was proved that cc(G) ≤ $$\frac{44}{27}$$ |E(G)| (≈ 1.6296|E(G)|) for loopless graphs with minimum degree at least 3. In this paper, we obtain results on integer 4-flows, which are used to find bounds for cc(G). We prove that if G has minimum degree at least 3 (loops being allowed), then cc(G)<1.6258|E(G)|. As a corollary, adding loops to vertices of degree 2, we obtain that cc(G)<1.6466|E(G)| for every bridgeless graph G with |V2(G)|≤ $$\frac{1}{30}$$ |E(G)|.