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1-factor and cycle covers of cubic graphs. (English) Zbl 1309.05108
Summary: Let $$G$$ be a bridgeless cubic graph. Consider a list of $$k$$ 1-factors of $$G$$. Let $$E_i$$ be the set of edges contained in precisely $$i$$ members of the $$k$$ 1-factors. Let $$\mu_k(G)$$ be the smallest $$|E_0|$$ over all lists of $$k$$ 1-factors of $$G$$. Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection.
Furthermore, if $$\mu_3(G)\neq0$$, then $$2\mu_3(G)$$ is an upper bound for the girth of $$G$$. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with $$\mu_4(G)=0$$ have a 4-cycle cover of length $$\frac{4}{3}|E(G)|$$ and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang [C.-Q. Zhang, Integer flows and cycle covers of graphs. New York, NY: Marcel Dekker (1996; Zbl 0866.05001)]. We also give a negative answer to a problem stated in [loc. cit.].

##### MSC:
 05C38 Paths and cycles
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##### References:
 [1] Alon, Covering multigraphs by simple circuits, SIAM J Algebraic Discrete Methods 6 pp 345– (1985) · Zbl 0581.05046 · doi:10.1137/0606035 [2] Bermond, Shortest coverings of graphs with cycles, J Combin Theory Ser B 35 pp 297– (1983) · Zbl 0559.05037 · doi:10.1016/0095-8956(83)90056-4 [3] Brinkmann, Generation and properties of snarks, J Combin Theory Ser B 103 pp 468– (2013) · Zbl 1301.05119 · doi:10.1016/j.jctb.2013.05.001 [4] Fan, Fulkerson’s conjecture and circuit covers, J Combin Theory Ser B 61 pp 133– (1994) · Zbl 0811.05053 · doi:10.1006/jctb.1994.1039 [5] J. L. Fouquet J. M. Vanherpe On the perfect matching index of bridgeless cubic graphs [6] Fulkerson, Blocking and antiblocking pairs of polyhedra, Math Program 1 pp 168– (1971) · Zbl 0254.90054 · doi:10.1007/BF01584085 [7] Häggkvist, Problem 443. Special case of the Fulkerson Conjecture, Discrete Math 307 pp 650– (2007) [8] X. Hou H.-J. Lai C.-Q. Zhang On matching coverings and cycle coverings [9] Jaeger, Conjecture 1 and 2, In: Combinatorics 79’ (M. Deza, I. G. Rosenberg, Eds.), Ann Disc Math 9 pp 305– (1980) [10] Kochol, Snarks without small cycles, J Combin Theory Ser B 67 pp 34– (1996) · Zbl 0855.05066 · doi:10.1006/jctb.1996.0032 [11] Máčajová, Short cycle covers of graphs and nowhere-zero flows, J Graph Theory 68 pp 340– (2011) · Zbl 1234.05138 · doi:10.1002/jgt.20563 [12] Máčajová, Technical reports in Informatics TR-2009-020, Faculty of Mathematics, Physics, and Informatics (2009) [13] Máčajová, Constructing hypohamiltonian snarks with cyclic connectivity 5 and 6, Electronic J Combin 14 (2007) [14] Mazzuoccolo, The equivalence of two conjectures of Berge and Fulkerson, J Graph Theory 68 pp 125– (2011) · Zbl 1230.05238 · doi:10.1002/jgt.20545 [15] Petersen, Die Theorie der regulären graphs, Acta Mathematica 15 pp 193– (1891) · JFM 23.0115.03 · doi:10.1007/BF02392606 [16] Seymour, On multicolourings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc London Math Soc 38 (3) pp 423– (1979) · Zbl 0411.05037 · doi:10.1112/plms/s3-38.3.423 [17] Steffen, Classifications and characterizations of snarks, Discrete Math 188 pp 183– (1998) · Zbl 0956.05089 · doi:10.1016/S0012-365X(97)00255-0 [18] Zhang, Integer flows and cycle covers of graphs (1997)
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