 
Summary: RAINBOWS IN THE HYPERCUBE
MARIA AXENOVICH, HEIKO HARBORTH,
ARNFRIED KEMNITZ,
MEINHARD M ¨OLLER, AND INGO SCHIERMEYER
November 4, 2004
Abstract. Let Qn be a hypercube of dimension n, that is, a graph whose vertices are
binary ntuples and two vertices are adjacent iff the corresponding ntuples differ in
exactly one position. An edge coloring of a graph H is called rainbow if no two edges
of H have the same color. Let f(G, H) be the largest number of colors such that there
exists an edge coloring of G with f(G, H) colors such that no subgraph isomorphic to
H is rainbow. In this paper we start the investigation of this antiRamsey problem by
providing bounds on f(Qn, Qk) which are asymptotically tight for k = 2 and by giving
some exact results.
1. Introduction
We consider finite and simple graphs. An edge coloring c : E(G) {1, 2, . . .} of a graph
G = (V (G), E(G)) is called rainbow if no two edges of G have the same color, that is, in a
rainbow coloring the edges are totally multicolored. Given a host graph G and a subgraph
H G, an edge coloring is called HantiRamsey iff every copy of H in G has at least two
edges of the same color. Denote by f(G, H) the maximum number of colors such that there
is no rainbow copy of H in some edge coloring of G with f(G, H) colors (which is of course
