Anti-Ramsey numbers of graphs with small connected components
Abstract
The anti-Ramsey number, AR(n,G), for a graph G and an integer n≥|V(G)|, is defined to be the minimal integer r such that in any edge-colouring of Kn by at least r colours there is a multicoloured copy of G, namely, a copy of G that each of its edges has a distinct colour. In this paper we determine, for large enough n, AR(n,L tP2) and AR(n,L kP3) for any large enough t and k, and a graph L satisfying some conditions. Consequently, we determine AR(n,G), for large enough n, where G is P3 tP2 for any t≥ 3, P4 tP2 and C3 tP2 for any t≥ 2, kP3 for any k≥ 3, tP2 kP3 for any t≥ 1, k≥ 2, and Pt+1 kP3 for any t≥ 3, k≥ 1. Furthermore, we obtain upper and lower bounds for AR(n,G), for large enough n, where G is Pk+1 tP2 and Ck tP2 for any k≥ 4, t≥ 1.
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