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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References
N. Alon and M. Tarsi: Covering multigraphs by simple circuits, SIAM J. Algebraic Discrete Methods 6 (1985), 345–350.
J. C. Bermond, B. Jackson and F. Jaeger: Shortest coverings of graphs with cycles, J. Combin. Theory Ser. B 35 (1983), 297–308.
P. A. Catlin: Double cycle covers and the Petersen graph, J. Graph Theory 13 (1989), 465–483.
G. Fan: Covering weighted graphs by even subgraphs, J. Combin. Theory Ser. B 49 (1990), 137–141.
G. Fan: Short cycle covers of cubic graphs, J. Graph Theory 18 (1994), 131–141.
G. Fan and A. Raspaud: Fulkerson’s Conjecture and circuit covers, J. Combinatorial Theory Ser. B 61 (1994), 133–138.
B. Jackson: Shortest circuit covers of cubic graphs, J. Combin. Theory Ser. B 60 (1994), 299–307.
F. Jaeger: Flows and generalized coloring theorems in graphs, J. Combin. Theory Ser. B 26 (1979), 205–216.
U. Jamshy, A. Raspaud and M. Tarsi: Short circuit covers for regular matroids with a nowhere zero 5-flow, J. Combin. Theory Ser. B 42 (1987), 354–357.
U. Jamshy and M. Tarsi: Short cycle covers and the cycle double cover conjecture, J. Combin. Theory Ser. B 56 (1992), 197–204.
T. Kaiser, D. Král, B. Lidický, P. Nejedlý and R. Šámal: Short cycle covers of graphs with minimum degree three, SIAM J. Discrete Math. 24 (2010), 330–355.
H. Lai and Z. Chen: Even subgraphs of a graph, Combinatorics, Graph Theory and Algorithms, New Issues Press, Kalamazoo (1999), 221–226.
P. D. Seymour: Sums of circuits, in: Graph Theory and Related Topics (J. A. Bondy and U. S. R. Murty, Eds.), 341–355, Academic Press, New York, 1979.
P. D. Seymour: Nowhere-zero 6-flows, J. Combin. Theory Ser. B 30 (1981), 130–135.
P. D. Seymour: On Tuttes extension of the four-colour problem, J. Combin. Theory Ser. B 31 (1981), 82–94.
G. Szekeres: Polyhedral decompositions of cubic graphs, Bull. Austral. Math. Soc. 8 (1973), 367–387.
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Research supported by NSFC Grant 11331003.