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)|.

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Correspondence to Genghua Fan.

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Research supported by NSFC Grant 11331003.

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Fan, G. Integer 4-Flows and Cycle Covers. Combinatorica 37, 1097–1112 (2017). https://doi.org/10.1007/s00493-016-3379-9

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Mathematics Subject Classification (2000)

  • 05C70
  • 05C38
  • 05C15