The anti-Ramsey numbers of C3 and C4 in complete r-partite graphs

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 C 3 and C 4 in complete r-partite graphs. For r ≥ 3 and n 1 ≥ n 2 ≥ ⋯ ≥ n r ≥ 1 , we determine ar ( K n 1 , n 2 , … , n r , { C 3 , C 4 } ) , ar ( K n 1 , n 2 , … , n r , C 3 ) and ar ( K n 1 , n 2 , … , n r , C 4 ) .