608 research outputs found

    Coloring squares of graphs with mad constraints

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    A proper vertex kk-coloring of a graph G=(V,E)G=(V,E) is an assignment c:V→{1,2,
,k}c:V\to \{1,2,\ldots,k\} of colors to the vertices of the graph such that no two adjacent vertices are associated with the same color. The square G2G^2 of a graph GG is the graph defined by V(G)=V(G2)V(G)=V(G^2) and uv∈E(G2)uv \in E(G^2) if and only if the distance between uu and vv is at most two. We denote by χ(G2)\chi(G^2) the chromatic number of G2G^2, which is the least integer kk such that a kk-coloring of G2G^2 exists. By definition, at least Δ(G)+1\Delta(G)+1 colors are needed for this goal, where Δ(G)\Delta(G) denotes the maximum degree of the graph GG. In this paper, we prove that the square of every graph GG with mad(G)<4\text{mad}(G)<4 and Δ(G)â©Ÿ8\Delta(G) \geqslant 8 is (3Δ(G)+1)(3\Delta(G)+1)-choosable and even correspondence-colorable. Furthermore, we show a family of 22-degenerate graphs GG with mad(G)<4\text{mad}(G)<4, arbitrarily large maximum degree, and χ(G2)â©Ÿ5Δ(G)2\chi(G^2)\geqslant \frac{5\Delta(G)}{2}, improving the result of Kim and Park.Comment: 14 pages, 4 figure

    Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

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    We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatoric

    Pushable chromatic number of graphs with degree constraints

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    International audiencePushable homomorphisms and the pushable chromatic number χp\chi_p of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph G→\overrightarrow{G}, we have χp(G→)≀χo(G→)≀2χp(G→)\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G}), where χo(G→)\chi_o(\overrightarrow{G}) denotes the oriented chromatic number of G→\overrightarrow{G}. This stands as the first general bounds on χp\chi_p. This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all Δ≄29\Delta \geq 29, we first prove that the maximum value of the pushable chromatic number of a connected oriented graph with maximum degree Δ\Delta lies between 2Δ2−12^{\frac{\Delta}{2}-1} and (Δ−3)⋅(Δ−1)⋅2Δ−1+2(\Delta-3) \cdot (\Delta-1) \cdot 2^{\Delta-1} + 2 which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when Δ≀3\Delta \leq 3, we then prove that the maximum value of the pushable chromatic number is~66 or~77. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~33 lies between~55 and~66. The former upper bound of~77 also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~66

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    2015 Symposium Brochure

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    Complete Issue 12, 1995

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    Novel Split-Based Approaches to Computing Phylogenetic Diversity and Planar Split Networks

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    EThOS - Electronic Theses Online ServiceGBUnited Kingdo
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