1,817 research outputs found

    The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths

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    The Ramsey number R(G,H)R(G, H) for a pair of graphs GG and HH is defined as the smallest integer nn such that, for any graph FF on nn vertices, either FF contains GG or F‾\overline{F} contains HH as a subgraph, where F‾\overline{F} denotes the complement of FF. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers R(K1+Ln,Pm)R(K_1+L_n, P_m) and R(K1+Ln,Cm)R(K_1+L_n, C_m) for some integers mm, nn, where LnL_n is a linear forest of order nn with at least one edge

    On path-quasar Ramsey numbers

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    Let G1G_1 and G2G_2 be two given graphs. The Ramsey number R(G1,G2)R(G_1,G_2) is the least integer rr such that for every graph GG on rr vertices, either GG contains a G1G_1 or G‾\overline{G} contains a G2G_2. Parsons gave a recursive formula to determine the values of R(Pn,K1,m)R(P_n,K_{1,m}), where PnP_n is a path on nn vertices and K1,mK_{1,m} is a star on m+1m+1 vertices. In this note, we first give an explicit formula for the path-star Ramsey numbers. Secondly, we study the Ramsey numbers R(Pn,K1∨Fm)R(P_n,K_1\vee F_m), where FmF_m is a linear forest on mm vertices. We determine the exact values of R(Pn,K1∨Fm)R(P_n,K_1\vee F_m) for the cases m≤nm\leq n and m≥2nm\geq 2n, and for the case that FmF_m has no odd component. Moreover, we give a lower bound and an upper bound for the case n+1≤m≤2n−1n+1\leq m\leq 2n-1 and FmF_m has at least one odd component.Comment: 7 page

    Small Ramsey Numbers

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    We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values

    Three results on cycle-wheel Ramsey numbers

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    Given two graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that, for any graph G of order N, either G1 is a subgraph of G, or G2 is a subgraph of the complement of G. We consider the case that G1 is a cycle and G2 is a (generalized) wheel. We expand the knowledge on exact values of Ramsey numbers in three directions: large cycles versus wheels of odd order; large wheels versus cycles of even order; and large cycles versus generalized odd wheels

    THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

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    and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

    Some Known Results and an Open Problem of Tree - Wheel Graph Ramsey Numbers

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    There are many famous problems on finding a regular substructure in a sufficiently large combinatorial structure, one of them i.e. Ramsey numbers. In this paper we list some known results and an open problem on graph Ramsey numbers. In the special cases, we list to determine graph Ramsey numbers for trees versus wheel

    Some Known Results and an Open Problem of Tree - Wheel Graph Ramsey Numbers

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    There are many famous problems on finding a regular substructure in a sufficiently large combinatorial structure, one of them i.e. Ramsey numbers. In this paper we list some known results and an open problem on graph Ramsey numbers. In the special cases, we list to determine graph Ramsey numbers for trees versus wheel
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