The Genus of Repeated Cartesian Products of Bipartite Graphsc)

With the aid of techniques developed by Edmonds, Ringel, and Youngs, it is shown that the genus of the cartesian product of the complete bipartite graph K2m,2m with itself is l + 8m2(m — 1). Furthermore, let ßi" be the graph K„,s and recursively define the cartesian product ßi," = ß?L x x Klfl for nä2. The genus of ß(„" is shown to be 1 + 2" " 3s"(sn—4), for all n, and í even ; or for n > 1, and s = 1 or 3. The graph ßi,1' is the 1-skeleton of the «-cube, and the formula for this case gives a result familiar in the literature. Analogous results are developed for repeated cartesian products of paths and of even cycles. Introduction. In this paper a graph G is a finite 1-complex. The genus y(G) of G is the minimum genus among the genera of all compact orientable 2-manifolds in which G can be imbedded. All 2-manifolds in this paper are assumed to be compact and orientable. There are very few families of graphs for which the genus has been determined; these include the complete graphs (Ringel and Youngs [7]), the complete bipartite graphs, (Ringel [5]), and some subfamilies of the family of complete tripartite graphs (see [6] and [8]). One of the first genus formulae was developed by Ringel [4] in 1955 (and independently by Beineke and Harary [1] in 1965) when he found that the genus of the M-cube Qn is given by: y(Qn) = l+2"-3(n-4), for n ^ 2. The «-cube can be defined as a repeated cartesian product: let Qx = K2, the complete graph on two vertices, and recursively define Qn = Qn _ x x K2 for n ^ 2. In general, given two graphs Gx and G2, with vertex sets V(GX), V(G2) and edge sets E(GX), E(G2) respectively, the cartesian product Gx x G2 is formed by taking V(GX x G2) ={(ux, u2) : ux e V(GX), u2 e V(G2)} and E(GX x G2)={[(ux, u2), (vx, v2)]: ux = vx Received by the editors October 6, 1969. AMS Subject Classifications. Primary 0550; Secondary 5475, 5510.