Distinguishing colorings of Cartesian products of complete graphs

We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph Ks,t which admits only the identity automorphism. In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs. The distinguishing number of a graph is the minimum number of colors needed to label the vertices so that the only color preserving automorphism is the identity. The distinguishing number was introduced by Albertson and Collins in [2] and a number of papers on this topic have been written recently. In this paper we determine values of c, s, t for which the Cartesian product of complete graphs of sizes s and t have an identity c coloring. In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs. For s ≤ t, the distinguishing number of the Cartesian product of complete graphs on s and t vertices, D(Ks2Kt) is either d(t+1)1/se or d(t+1)1/se+1 and it is the smaller value for large enough t. In almost all cases it can be determined directly which value holds. In a few remaining cases the value can be determined by a simple recursion. Our original version of this paper [3] was motivated by a problem of Harary and titled ‘Edge colored complete bipartite graphs with trivial automorphism groups’. We later discovered the connection to the distinguishing number. The current version has a final added section making the connection to distinguishing numbers. Thus the rest of paper, except the final section where we make the connection to distinguishing numbers is the original version written in terms of identity edge colorings of complete bipartite graphs. ∗Department of Mathematics, California State University, Fresno, Fresno, CA 93740, email: [email protected] †Department of Mathematics, Lehigh University, Bethlehem, PA 18015, email: [email protected]