53 research outputs found

    metodologi terapan

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    Inside-Out Polytopes

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    We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat

    Group Irregularity Strength of Connected Graphs

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    We investigate the group irregularity strength (sg(G)s_g(G)) of graphs, i.e. the smallest value of ss such that taking any Abelian group \gr of order ss, there exists a function f:E(G)\rightarrow \gr such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph GG of order at least 3, sg(G)=ns_g(G)=n if n≠4k+2n\neq 4k+2 and sg(G)≤n+1s_g(G)\leq n+1 otherwise, except the case of some infinite family of stars

    Group Sum Chromatic Number of Graphs

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    We investigate the \textit{group sum chromatic number} (\gchi(G)) of graphs, i.e. the smallest value ss such that taking any Abelian group \gr of order ss, there exists a function f:E(G)\rightarrow \gr such that the sums of edge labels properly colour the vertices. It is known that \gchi(G)\in\{\chi(G),\chi(G)+1\} for any graph GG with no component of order less than 33 and we characterize the graphs for which \gchi(G)=\chi(G).Comment: Accepted for publication in European Journal of Combinatorics, Elsevie
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