Totally Multicolored diamonds

Let G be a graph of order p. For every n ≥ p let f ( n , G ) be the minimum integer k such that for every edge-coloring of the complete graph of order n which uses exactly k colors, there is at least one copy of G all whose edges have different colors. Let F be a set of graphs. For every n ≥ 3 let ext ( n , F ) be the maximum number of edges of a graph on n vertices with no subgraph isomorphic to an element of F . Here we study the relation between f ( n , G ) and ext ( n , C ( G ) ) when G is a graph with chromatic number 3 obtained by adding an edge (a chord) to a cycle, and C ( G ) is the set of cycles which are subgraphs of G. In particular, an upperbound and a lowerbound of f ( n , G ) are given; and in the case when G is the diamond ( C 4 with a chord), we prove that the supremum and infimum limits of f ( n , G ) n n are bounded by 2 3 and 1 2 2 respectively, and we conjecture that for every n ≥ 4 , f ( n , G ) = ext ( n , { C 3 , C 4 } ) + 2 .