30,131 research outputs found

    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

    Impartial coloring games

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    Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of k>0k > 0 colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function

    Precoloring co-Meyniel graphs

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    The pre-coloring extension problem consists, given a graph GG and a subset of nodes to which some colors are already assigned, in finding a coloring of GG with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. We answer a question of Hujter and Tuza by showing that ``PrExt perfect'' graphs are exactly the co-Meyniel graphs, which also generalizes results of Hujter and Tuza and of Hertz. Moreover we show that, given a co-Meyniel graph, the corresponding contracted graph belongs to a restricted class of perfect graphs (``co-Artemis'' graphs, which are ``co-perfectly contractile'' graphs), whose perfectness is easier to establish than the strong perfect graph theorem. However, the polynomiality of our algorithm still depends on the ellipsoid method for coloring perfect graphs
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