Distinguishing threshold of graphs

Abstract

AbstractA vertex coloring of a graph is called distinguishing if no nonidentity automorphisms of can preserve it. The distinguishing number of , denoted by , is the minimum number of colors required for such a coloring, and the distinguishing threshold of , denoted by , is the minimum number such that every ‐coloring of is distinguishing. As an alternative definition, is one more than the maximum number of cycles in the cycle decomposition of automorphisms of . In this paper, we characterize when is disconnected. Afterwards, we prove that, although for every positive integer there are infinitely many graphs whose distinguishing thresholds are equal to , we have if and only if . Moreover, we show that if , then either is isomorphic to one of the four graphs on three vertices or it is of order , where is a prime number. Furthermore, we prove that if and only if is asymmetric, or . Finally, we consider all generalized Johnson graphs, , which are the graphs on all ‐subsets of where two vertices and are adjacent if . After studying their automorphism groups and distinguishing numbers, we calculate their distinguishing thresholds as , unless and in which case we have .