Anti-Ramsey numbers for disjoint copies of graphs

Abstract

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph \(G\) and a positive integer \(n\), the anti-Ramsey number \(ar(n,G)\) is the maximum number of colors in an edge-coloring of \(K_n\) with no rainbow copy of \(H\). Anti-Ramsey numbers were introduced by Erdȍs, Simonovits and Sos and studied in numerous papers. Let \(G\) be a graph with anti-Ramsey number \(ar(n,G)\). In this paper we show the lower bound for \(ar(n,pG)\), where \(pG\) denotes \(p\) vertex-disjoint copies of \(G\). Moreover, we prove that in some special cases this bound is sharp.