Local rainbow colorings

Abstract

Given a graph H, we denote by C(n, H) the minimum number k such that the following holds. There are n colorings of E(K n) with k-colors, each associated with one of the vertices of K n , such that for every copy T of H in K n , at least one of the colorings that are associated with V (T) assigns distinct colors to all the edges of E(T). We characterize the set of all graphs H for which C(n, H) is bounded by some absolute constant c(H), prove a general upper bound and obtain lower and upper bounds for several graphs of special interest. A special case of our results partially answers an extremal question of Karchmer and Wigderson motivated by the investigation of the computational power of span programs.