The Ramsey Numbers of Paths Versus Fans


Salman, A.N.M. and Broersma, H.J. (2003) The Ramsey Numbers of Paths Versus Fans. In: Hajo Broersma & Ulrich Faigle & Johann Hurink & Stefan Pickl & Gerhard Woeginger (Eds.), 2nd Cologne-Twente Workshop on Graphs and Combinatorial Optimization. Electronic Notes in Discrete Mathematics, 13 . Elsevier, pp. 103-107.

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Abstract:For two given graphs G and H, the Ramsey number R(G,H) is the smallest positive integer p such that for every graph F on p vertices the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we study the Ramsey numbers R(Pn,Fm), where Pn is a path on n vertices and Fm is the graph obtained from m disjoint triangles by identifying precisely one vertex of every triangle (Fm is the join of K1 and mK2). We determine exact values for R(Pn,Fm) for the following values of n and m: n = 1,2 or 3 and m ≥ 2; n ≥ 4 and 2 ≤ m ≤ (n + 1)/2; n ≥ 7 and m = n − 1 or m = n; n ≥ 8 and (k · n − 2k + 1)/2 ≤ m ≤ (k · n − k + 2)/2 with 3 ≤ k ≤ n − 5; n = 4,5 or 6 and m ≥ n − 1; n ≥ 7 and m ≥ (n − 3)2/2.
Item Type:Book Section
Copyright:© 2003 Elsevier
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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