Anti-Ramsey number of intersecting cliques

Abstract

An edge-colored graph is called a rainbow graph if all its edges have distinct colors. The anti-Ramsey number a r ( n , G ) , for a graph G and a positive integer n , is defined as the minimum number of colors r such that every exact r -edge-coloring of the complete graph K n contains at least one rainbow copy of G . A ( k , r ) -fan graph, denoted by F k , r , is a graph composed of k cliques each with r vertices, intersecting at exactly one common vertex. In this paper, we determine a r ( n , F k + 1 , r ) for n ≥ 256 r 16 ( k + 1 ) 5 , k ≥ 1 , and r ≥ 4 .