Identity orientation of complete bipartite graphs

An identity orientation of a graph G=(V,E) is an orientation of some of the edges of E such that the resulting partially oriented graph has no automorphism other than the identity. We show that the complete bipartite graph Ks,t, with s⩽t, does not have an identity orientation if t⩾3s-⌈log3(s-1)⌉. We also show that if (r+1)(r+2)⩾2s then Ks,3s-r does have an identity orientation. These results improve the previous bounds obtained by Harary and Jacobson (Discuss. Math. - Graph Theory 21 (2001) 158). We use these results to determine exactly the values of t for which an identity orientation of Ks,t exists for 2⩽s⩽17.