Distinguishing colorings of Cartesian products of complete graphs

Abstract

We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph K s,t which admits only the identity automorphism. In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs.

Improving upper bounds for the distinguishing index

“…For a 2-connected planar graph G, the distinguishing index may attain 1 + ∆(G) as it is shown by the complete bipartite graph K 2,q with q = r 2 for a positive integer r. In this case, D (K 2,q ) = r + 1 as it follows from the result obtained independently by Fisher and Isaak [11] and by Imrich, Jerebic and Klavžar [14]. They proved the following theorem.…”

The chromatic distinguishing index of certain graphs

“…In fact, the rows of A are distinct, and since the number of label t + i − 1(mod s) in the ith column is one and in the jth column, j = i, is zero, so the columns have distinct degrees. Hence χ ′ D (K s,t ) = s. Before we prove the next result, we need the following preliminaries: By the result obtained by Fisher and Isaak [4] and independently by Imrich, Jerebic and Klavžar [7] the distinguishing index of complete bipartite graphs is as follows.…”

“…Here, we want to obtain the distinguishing chromatic index of complete bipartite graphs. Before we obtain the distinguishing chromatic index of complete bipartite graphs we need the following information of [4]: A labeling with c labels of the edges of a complete bipartite graph K s,t having parts X of size s and Y of size t corresponds to a t × s matrix with entries from {1, . .…”

The determining number of a Cartesian product

“…In general, the structure of Cartesian products makes matrices particularly useful for studying questions involving their automorphisms. See [7,16], and [13] for further examples of matrix techniques used to find determining sets or distinguishing labelings. …”