RAINBOWS IN THE HYPERCUBE MARIA AXENOVICH, HEIKO HARBORTH, ARNFRIED KEMNITZ, MEINHARD M ¨ OLLER, AND INGO SCHIERMEYER November 4, 2004 Abstract. Let Q n be a hypercube of dimension n, that is, a graph whose vertices are binary n-tuples and two vertices are adjacent iff the corresponding n-tuples 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 anti-Ramsey problem by providing bounds on f (Q n ,Q k ) 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 H -anti-Ramsey 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 the maximum number of colors to be used in an H -anti-Ramsey coloring of G). Equivalently, any edge coloring of G with at least rb(G, H )= f (G, H ) + 1 colors contains a rainbow copy of H . The number rb(G, H ) will be called rainbow number and f (G, H ) anti-Ramsey number of graphs G and H . The function f (G, H ) was introduced by Erd˝ os, Simonovits and S´ os [4]. They determined the asymptotic behavior of f (K n ,H ) for graphs H such that the deletion of any edge results in a graph with chromatic number at least 3 showing that this function is closely related to the extremal function ex(n, H ) which gives the maximum number of edges in a subgraph of K n containing no copy of a graph H . See also [7] for exact results. Anti-Ramsey problems drew a lot of attention in the past few years. Most of them are investi- gating f (K n ,H ), that is, when the edge-colored graph is complete (see, e.g., [6], [8], [9], [10]). There is a number of results on anti-Ramsey numbers under different local constraints and on anti-Ramsey numbers of the structures other than graphs, such as posets, integers and so on. 2000 Mathematics Subject Classification: 05C15. Keywords: Edge coloring, rainbow coloring, anti-Ramsey coloring, hypercube. 1