26,027 research outputs found

    On Ramsey numbers of complete graphs with dropped stars

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    Let r(G,H)r(G,H) be the smallest integer NN such that for any 22-coloring (say, red and blue) of the edges of K_nK\_n, nNn\geqslant N, there is either a red copy of GG or a blue copy of HH. Let K_nK_1,sK\_n-K\_{1,s} be the complete graph on nn vertices from which the edges of K_1,sK\_{1,s} are dropped. In this note we present exact values for r(K_mK_1,1,K_nK_1,s)r(K\_m-K\_{1,1},K\_n-K\_{1,s}) and new upper bounds for r(K_m,K_nK_1,s)r(K\_m,K\_n-K\_{1,s}) in numerous cases. We also present some results for the Ramsey number of Wheels versus K_nK_1,sK\_n-K\_{1,s}.Comment: 9 pages ; 1 table in Discrete Applied Mathematics, Elsevier, 201

    On the Geometric Ramsey Number of Outerplanar Graphs

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    We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n2n vertices are bounded by O(n3)O(n^{3}) and O(n10)O(n^{10}), in the convex and general case, respectively. We then apply similar methods to prove an nO(log(n))n^{O(\log(n))} upper bound on the Ramsey number of a path with nn ordered vertices.Comment: 15 pages, 7 figure
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