Abstract
Given k directed graphs G 1,…, G k the Ramsey number R( G 1,…, G k ) is the smallest integer n such that for any partition ( U 1,…, U k ) of the arcs of the complete symmetric directed graph K n, there exists an integer i such that the partial graph generated by U 1 contains G 1 as a subgraph. In the article we give a necessary and sufficient condition for the existence of Ramsey numbers, and, when they exist an upper bound function. We also give exact values for some classes of graphs. Our main result is: R( P n,…. P nk-1, G) = n 1…n k-1 (p-1) + 1 , where G is a hamltonian directed graph with p vertices and P n i denotes the directed path of length n t
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