165 research outputs found

    Perfect Matchings in Claw-free Cubic Graphs

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    Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.Comment: 6 pages, 2 figure

    Normal 6-edge-colorings of some bridgeless cubic graphs

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    In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal kk-edge-coloring of a cubic graph is an edge-coloring with kk colors such that each edge of the graph is normal. We denote by χN′(G)\chi'_{N}(G) the smallest kk, for which GG admits a normal kk-edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. It is known that proving χN′(G)≤5\chi'_{N}(G)\leq 5 for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover, Jaeger was able to show that it implies classical conjectures like Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors were able to show that any simple cubic graph admits a normal 77-edge-coloring, and this result is best possible. In the present paper, we show that any claw-free bridgeless cubic graph, permutation snark, tree-like snark admits a normal 66-edge-coloring. Finally, we show that any bridgeless cubic graph GG admits a 66-edge-coloring such that at least 79⋅∣E∣\frac{7}{9}\cdot |E| edges of GG are normal.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with arXiv:1804.0944

    A note on 2--bisections of claw--free cubic graphs

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    A \emph{kk--bisection} of a bridgeless cubic graph GG is a 22--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most kk. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a 22--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs
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