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Local Rainbow Colorings Ido Ben-Eliezer

Summary: Local Rainbow Colorings
Noga Alon
Ido Ben-Eliezer
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 k-colors, 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.


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics