Abstract
A subgraph of an edge-colored graph is rainbow, if all of its edges have different colors. For a graph G and a family H of graphs, the anti-Ramsey number ar(G,H) is the maximum number k such that there exists an edge-coloring of G with exactly k colors without rainbow copy of any graph in H. In this paper, we study the anti-Ramsey numbers of C3 and C4 in complete r-partite graphs. For r≥3 and n1≥n2≥⋯≥nr≥1, we determine ar(Kn1,n2,…,nr,{C3,C4}),ar(Kn1,n2,…,nr,C3) and ar(Kn1,n2,…,nr,C4).
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