Number of distinguishing colorings and partitions

Abstract

A vertex coloring of a graph G is called distinguishing (or symmetry breaking) if no non-identity automorphism of G preserves it, and the distinguishing number, shown by D(G), is the smallest number of colors required for such a coloring. This paper is about counting non-equivalent distinguishing colorings of graphs with k colors. A parameter, namely Φk(G), which is the number of non-equivalent distinguishing colorings of a graph G with at most k colors, is shown here to have an application in calculating the distinguishing number of the lexicographic product and the X-join of graphs. We study this index (and some other similar indices) which is generally difficult to calculate. Then, we show that if one knows the distinguishing threshold of a graph G, which is the smallest number of colors θ(G) so that, for k≥θ(G), every k-coloring of G is distinguishing, then, in some special cases, counting the number of distinguishing colorings with k colors is very easy. We calculate θ(G) for some classes of graphs including the Kneser graph K(n,2). We then turn to vertex partitioning by studying the distinguishing coloring partition of a graph G; a partition of vertices of G which induces a distinguishing coloring for G. There, we introduce Ψk(G) as the number of non-equivalent distinguishing coloring partitions with at most k cells, which is a generalization to its distinguishing coloring counterpart.