 
Summary: Local Rainbow Colorings
Noga Alon
Ido BenEliezer
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(Kn) with kcolors, each associated with one of the vertices of Kn, such
that for every copy T of H in Kn, 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.
1 Introduction
Consider the following question, motivated by an extremal problem suggested by Karchmer and
Wigderson, see [6]. Given a fixed graph H, let C(n, H) denote the minimum number k such that
there is a set of n colorings {fv : E(Kn) [k] : v V (Kn)}, with the following property. For every
copy T of H in Kn, there is a vertex u V (T) so that fu is a rainbow coloring of E(T), that is, no
two edges of T get the same color by fu. A set of colorings that satisfies this condition is called an
(n, H)local coloring. Determine or estimate the function C(n, H). Note that each coloring in the set
above does not have to be a proper edge coloring.
