190 research outputs found

    Full Orientability of the Square of a Cycle

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    Let D be an acyclic orientation of a simple graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define m and M to be the minimum and the maximum number of d(D) over all acyclic orientations D of G. We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n is fully orientable except n=6.Comment: 7 pages, accepted by Ars Combinatoria on May 26, 201

    Acyclic list edge coloring of outerplanar graphs

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    AbstractAn acyclic list edge coloring of a graph G is a proper list edge coloring such that no bichromatic cycles are produced. In this paper, we prove that an outerplanar graph G with maximum degree Ξ”β‰₯5 has the acyclic list edge chromatic number equal to Ξ”

    The strong chromatic index of 1-planar graphs

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    The chromatic index Ο‡β€²(G)\chi'(G) of a graph GG is the smallest kk for which GG admits an edge kk-coloring such that any two adjacent edges have distinct colors. The strong chromatic index Ο‡sβ€²(G)\chi'_s(G) of GG is the smallest kk such that GG has a proper edge kk-coloring with the condition that any two edges at distance at most 2 receive distinct colors. A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every graph GG with maximum average degree dΛ‰(G)\bar{d}(G) has Ο‡sβ€²(G)≀(2dΛ‰(G)βˆ’1)Ο‡β€²(G)\chi'_{s}(G)\le (2\bar{d}(G)-1)\chi'(G). As a corollary, we prove that every 1-planar graph GG with maximum degree Ξ”\Delta has Ο‡sβ€²(G)≀14Ξ”\chi'_{\rm s}(G)\le 14\Delta, which improves a result, due to Bensmail et al., which says that Ο‡sβ€²(G)≀24Ξ”\chi'_{\rm s}(G)\le 24\Delta if Ξ”β‰₯56\Delta\ge 56

    A note on the adjacent vertex distinguishing total chromatic number of graphs

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    AbstractAn adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have different sets of colors. The minimum number of colors needed for such a total coloring of G is denoted by Ο‡aβ€³(G). In this note, we show that Ο‡aβ€³(G)≀2Ξ” for any graph G with maximum degree Ξ”β‰₯3
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