2 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)⩾8\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)⩾5Δ(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
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