Number of Distinguishing Colorings and Partitions

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 $\Phi_k (G)$, 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 $\theta(G)$ so that, for $k\geq \theta(G)$, 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 $\theta(G)$ 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 $\Psi_k (G)$ as the number of non-equivalent distinguishing coloring partitions with at most $k$ cells, which is a generalization to its distinguishing coloring counterpart.