论文信息 - A note on Berge-Fulkerson coloring

A note on Berge-Fulkerson coloring

The Berge-Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings. In this paper, a useful technical lemma is proved that a cubic graphGadmits a Berge-Fulkerson coloring if and only if the graphGcontains a pair of edge-disjoint matchingsM"1andM"2 such that (i) M"[email protected]?M"2induces a 2-regular subgraph ofG and (ii) the suppressed [email protected]?M"[email protected]?, the graph obtained [email protected]?M"iby suppressing all degree-2-vertices, is 3-edge-colorable for eachi=1,2. This lemma is further applied in the verification of Berge-Fulkerson Conjecture for some families of non-3-edge-colorable cubic graphs (such as, Goldberg snarks, flower snarks).

[1] Jack Edmonds,et al. Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[2] J. A. Bondy,et al. Graph Theory with Applications , 1978 .

[3] Michael Tarsi,et al. Short cycle covers and the cycle double cover conjecture , 1992, J. Comb. Theory, Ser. B.

[4] Cun-Quan Zhang. Integer Flows and Cycle Covers of Graphs , 1997 .

[5] Mark K. Goldberg,et al. Construction of class 2 graphs with maximum vertex degree 3 , 1981, J. Comb. Theory, Ser. B.

[6] Bill Jackson,et al. Shortest coverings of graphs with cycles , 1983, J. Comb. Theory, Ser. B.

[7] R. Isaacs. Infinite Families of Nontrivial Trivalent Graphs Which are not Tait Colorable , 1975 .

[8] G. Szekeres,et al. Polyhedral decompositions of cubic graphs , 1973, Bulletin of the Australian Mathematical Society.

[9] D. R. Fulkerson,et al. Blocking and anti-blocking pairs of polyhedra , 1971, Math. Program..

[10] Amanda G. Chetwynd,et al. The rise and fall of the critical graph conjecture , 1983, J. Graph Theory.

[11] Elwood S. Buffa,et al. Graph Theory with Applications , 1977 .

[12] P. D. Seymour,et al. On Multi‐Colourings of Cubic Graphs, and Conjectures of Fulkerson and Tutte , 1979 .

[13] Genghua Fan. Covering Graphs by Cycles , 1992, SIAM J. Discret. Math..

[14] André Raspaud,et al. Fulkerson's Conjecture and Circuit Covers , 1994, J. Comb. Theory, Ser. B.