Discrete Applied Mathematics
The Ramsey Numbers of Large cycles Versus Odd Wheels
Graphs and Combinatorics
Journal of Graph Theory
Discrete Applied Mathematics - Special issue: 2nd cologne/twente workshop on graphs and combinatorial optimization (CTW 2003)
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Given two graphs G"1 and G"2, denote by G"1*G"2 the graph obtained from G"1@?G"2 by joining all the vertices of G"1 to the vertices of G"2. The Ramsey number R(G"1,G"2) is the smallest positive integer n such that every graph G of order n contains a copy of G"1 or its complement G^c contains a copy of G"2. It is shown that the Ramsey number of a book B"m=K"2*K"m^c versus a cycle C"n of order n satisfies R(B"m,C"n)=2n-1 for n(6m+7)/4 which improves a result of Faudree et al., and the Ramsey number of a cycle C"n versus a wheel W"m=K"1*C"m satisfies R(C"n,W"m)=2n-1 for even m and n=3m/2+1 and R(C"n,W"m)=3n-2 for odd m1 andn=3m/2+1 or nmax{m+1,70} or n=max{m,83} which improves a result of Surahmat et al. and also confirms their conjecture for large n. As consequences, Ramsey numbers of other sparse graphs are also obtained.