Rainbow coloring the cube

R. J. Faudree DEPARTMENT OF MATHEMATICAL SCIENCES MEMPHIS STATE UNIVERSITY MEMPHIS, TENNESSEE A. Gyarfas COMPUTER AND AUTOMATION INSTITUTE HUNGARIAN ACADEMY OF SCIENCES BUDAPEST, HUNGARY L. Lesniak DEPARTMENT OF MATH AND COMPUTER SCIENCE DREW UNIVERSITY MADISON, NEW JERSEY R. H. Schelp DEPARTMENT OF MATHEMATICAL SCIENCES MEMPHIS STATE UNIVERSITY MEMPHIS, TENNESSEE We prove that for d ~ 4, d * 5, the edges of the d-dimensional cube can be colored by d colors so that all quadrangles have four distinct colors. © 1993 John Wiley & Sons, Inc. At the recent 23rd Southeastern Conference on Graph Theory, Combinatorics, and Computing, Puhua Guan asked the following question: Is it possible to color the edges of the d-dimensional cube Qd with d colors so that all quadrangles of Qd are colored with four distinct colors? This makes sense only for d ~ 4 and Guan mentioned that he has constructed such a coloring for d = 4. In this article we give an affirmative answer to this question, except for d = 5, where the required coloring does not exist. We call an edge-coloring of a graph G a rainbow coloring if the edges of every quadrangle ( C4 in what follows) of G are colored with distinct colors. Let rb (G) denote the minimum number of colors in a rainbow coloring of G. Notice that rb(G) = 1 if G has no quadrangles, otherwise rb(G) ~ 4. Rainbow colorings seem particularly interesting for graphs having the following property: any two incident edges are in a quadrangle of the graph. Journal of Graph Theory, Vol. 17, No. 5, 607-612 {1993) © 1993 John Wiley & Sons, Inc. CCC 0364-9024/93/050607-06 608 JOURNAL OF GRAPH THEORY In this case, rainbow colorings are automatically proper edge colorings in the usual sense, i.e., each color class is the union of disjoint edges. Since this property is preserved under taking Cartesian products of graphs, it seems natural to study rb(G X H) in general. Although we focus our attention on Qd, some lemmas are used that point to the more general setting. Rainbow colorings are also related to total colorings. A coloring of edges and vertices (elements) of a graph is total if both edge and vertex colorings are proper and two elements of the same color are not incident. In what follows, c (x) and c (x, y) will be used to denote the color of a vertex x and edge xy, respectively. Theorem 1. If d :2: 4, d =I= 5 then rb(Qd) = d. Corollary. If d :2: 3, d =I= 4, there exists a total (d + I)-coloring of Qd, which is also a rainbow coloring. Proof. Let x be a rainbow (d + I)-coloring of Qd+I from Theorem 1. Consider Qd+ 1 as two disjoint copies of Qd with a factor between them. On one of these copies x induces a rainbow (d + I)-coloring and the colors of the factor edges give a proper vertex coloring on their end points in the copy of Qd in question. It is immediate that this coloring is total on Qd. 1 Perhaps at this point it is useful to remark that it is easy to construct directly a total (d + I)-coloring of Qd for d :2: 3 (without the additional rainbow property). This can be done by induction on d as follows. To anchor the induction, take a total 4-coloring of Q3 (see Figure 1). For the inductive step, take two disjoint copies of Qd, say A and B. Join the corresponding vertices of A and B by a factor XiYi, i = I, 2, ... , 2d. Set c(xi, Yi) = d + 2. Select any permutation II on the set {I, 2, ... , d + I} of colors that has no fixed point. Take a total (d + I)-coloring on A using colors I, 2, ... , d + I (induction) and permute colors on B as follows: