28 research outputs found

    Class two 1-planar graphs with maximum degree six or seven

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    A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.Comment: 3 pages, 2 figure

    Total coloring of 1-toroidal graphs of maximum degree at least 11 and no adjacent triangles

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    A {\em total coloring} of a graph GG is an assignment of colors to the vertices and the edges of GG such that every pair of adjacent/incident elements receive distinct colors. The {\em total chromatic number} of a graph GG, denoted by \chiup''(G), is the minimum number of colors in a total coloring of GG. The well-known Total Coloring Conjecture (TCC) says that every graph with maximum degree Δ\Delta admits a total coloring with at most Δ+2\Delta + 2 colors. A graph is {\em 11-toroidal} if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of 11-toroidal graphs, and prove that the TCC holds for the 11-toroidal graphs with maximum degree at least~1111 and some restrictions on the triangles. Consequently, if GG is a 11-toroidal graph with maximum degree Δ\Delta at least~1111 and without adjacent triangles, then GG admits a total coloring with at most Δ+2\Delta + 2 colors.Comment: 10 page

    Cyclic Coloring of Plane Graphs with Maximum Face Size 16 and 17

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    Plummer and Toft conjectured in 1987 that the vertices of every 3-connected plane graph with maximum face size D can be colored using at most D+2 colors in such a way that no face is incident with two vertices of the same color. The conjecture has been proven for D=3, D=4 and D>=18. We prove the conjecture for D=16 and D=17

    Third case of the Cyclic Coloring Conjecture

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    The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring Conjecture has been proven only for two values of D: the case D=3 is equivalent to the Four Color Theorem and the case D=4 is equivalent to Borodin's Six Color Theorem, which says that every graph that can be drawn in the plane with each edge crossed by at most one other edge is 6-colorable. We prove the case D=6 of the conjecture
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