Edge colored complete bipartite graphs with trivial automorphism groups

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, we show that for a given s with a few small exceptions, there is such a coloring with c colors if and only r ≤ t ≤ cs − r where r = blogc(s − 1)c + 1 if s ≤ c or if s ≥ c + 1 and s ≥ c1+blogc(s−1)c − blogcblogc(s− 1)cc and that r = blogc(s− 1)c+ 2 if s ≤ c1+blogc(s−1)c − blogcblogc(s− 1)cc − 2. When s = c1+blogc(s−1)c − blogcblogc(s− 1)cc−1 then r will be one of these two values and we can determine which recursively. Harary and Jacobson [1] examined the minimum number of edges that need to be oriented so that the resulting mixed graph has the trivial automorphism group and determined some values of s and t for which this number exists for the complete bipartite graph Ks,t. These are values for which there is a mixed graph resulting from orienting some of the edges with only the trivial automorphism. Such an orientation is called an identity orientation. Harary and Ranjan [2] determined further bounds on when Ks,t has an identity orientation. They showed that Ks,t does not have an identity orientation for t ≤ blog3(s − 1)c or t ≥ 3 − blog3(s − 1)c and that it does have an identity orientation for d√2s + 3/2e ≤ t ≤ 3 − d√2s + 3/2e. In addition they determined exact values when 2 ≤ s ≤ 17. We will show that the first bound is nearly correct. Observe that a partial orientation of a complete bipartite graph with parts X and Y has three types of edges: unoriented, oriented from X to Y , and oriented from Y to X. We can more generally think of coloring the edges with some number c of colors. ∗Department of Mathematics, California State University, Fresno, Fresno, CA 93740, email: [email protected] †Department of Mathematics, Lehigh University, Bethlehem, PA 18015, email: [email protected]