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European Journal of Combinatorics, 1982 In the first section of this paper it is shown that the bipartite Ramsey number br(m, n) satisfies br(m, n)⩽2m(n−1)+1. (Beineke and Schwenk [1] conjectured br(m, n) = 2m(n−1) + 1 but Irving [17] showed that equality does not always hold.) The second section gives a conjecture for the Ramsey numbers of the complete graphs, and lastly the numbers of ...
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Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths
Discussiones Mathematicae Graph Theory, 2015 The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H.
Hook Jonelle
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On the restricted size Ramsey number for P3 versus dense connected graphs
Electronic Journal of Graph Theory and Applications, 2020 Let F, G and H be simple graphs. A graph F is said a (G,H)-arrowing graph if in any red-blue coloring of edges of F we can find a red G or a blue H. The size Ramsey number of G and H, ŕ(G,H), is the minimum size of F.
Denny Riama Silaban +2 more
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RAMSEY NUMBERS FOR TREES [PDF]
Bulletin of the Australian Mathematical Society, 2012 AbstractForn≥5, letT′ndenote the unique tree onnvertices with Δ(T′n)=n−2, and letT*n=(V,E) be the tree onnvertices withV={v0,v1,…,vn−1} andE={v0v1,…,v0vn−3,vn−3vn−2,vn−2vn−1}. In this paper, we evaluate the Ramsey numbersr(Gm,T′n) andr(Gm,T*n) , whereGmis a connected graph of orderm.
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Diagonal Ramsey numbers in multipartite graphs related to stars
Electronic Journal of Graph Theory and Applications, 2022 : Let the star on n vertices, namely K1, n − 1 be denoted by Sn. If every two coloring of the edges of a complete balanced multipartite graph Kj × s there is a copy of Sn in the first color or a copy of Sm in the second color, then we will say Kj × s ...
Chula Janak Jayawardene
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Discrete Mathematics, 2002 Let \(TT_n\) denote the transitive tournament on \(n\) vertices and let \(D_1,D_2,\dots, D_k\) be acyclic digraphs. The ordered Ramsey number \(\rho(D_1,D_2,\dots, D_k)\) is the smallest integer \(n\) so that any \(k\)-coloring of the edges of \(TT_n\) admits, for some index \(i\), a copy of \(D_i\) with all of its edges colored with the \(i\)th color.
Sheshayya A. Choudum, B. Ponnusamy
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Ramsey and Gallai-Ramsey numbers for forests
Discussiones Mathematicae Graph TheorySummary: Given two non-empty graphs \(G,H\) and a positive integer \(k\), the Gallai-Ramsey number \(\operatorname{gr}_k(G:H)\) is defined as the minimum integer \(N\) such that for all \(n\geq N\), every \(k\)-edge-coloring of \(K_n\) contains either a rainbow copy of \(G\) or a monochromatic copy of \(H\).
Yujia Gao +3 more
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European Journal of Combinatorics, 1997 For positive integers \(s\) and \(t\), the matroid Ramsey number \(n(s,t)\) is the least positive integer \(n\) such that every connected matroid with at least \(n\) elements has either a circuit with at least \(s\) elements or a cocircuit with at least \(t\) elements.
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Discrete Mathematics, 2002 The induced Ramsey number \(\text{IR}(G,H)\) is the smallest integer \(n\) for which there exists a graph \(F\) on \(n\) vertices such that any 2-colouring of its edges in red and blue produces either or both of a copy of a graph \(G\) induced in \(F\) with all of its edges red, or a copy of a graph \(H\) induced in \(F\) with all of its edges blue.
Izolda Gorgol, Tomasz Luczak 0001
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Discrete Mathematics, 1999 The linear arboricity of a graph \(G\) is the least integer \(m\) such that the vertex set \(V(G)\) can be partitioned into \(m\) sets, each inducing a linear forest, i.e. a forest in which every component is a path. Let \(H\) be a connected graph on \(n\) vertices such that \(V(H)\) can be covered by \(t\) (and no fewer) vertex-disjoint paths in the ...
Baogen Xu, Zhongfu Zhang
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