The Distinguishing Index of Mycielskian Graphs

Abstract

The distinguishing index gives a measure of symmetry in a graph. Given a graph $G$ with no $K_2$ component, a distinguishing edge coloring is a coloring of the edges of $G$ such that no non-trivial automorphism preserves the edge coloring. The distinguishing index, denoted $\operatorname{Dist^{\prime}}(G)$, is the smallest number of colors needed for a distinguishing edge coloring. The Mycielskian of a graph $G$, denoted $\mu(G)$, is an extension of $G$ introduced by Mycielski in 1955. In 2020, Alikhani and Soltani conjectured a relationship between $operatorname{Dist^{\prime}}(G)$ and $operatorname{Dist^{\prime}}(\mu(G))$. We prove that for all graphs $G$ with at least 3 vertices, no connected $K_2$ component, and at most one isolated vertex, $\operatorname{Dist^{\prime}}(\mu(G)) \le \operatorname{Dist^{\prime}}(G)$, exceeding their conjecture. We also prove analogous results about generalized Mycielskian graphs. Together with the work in 2022 of Boutin, Cockburn, Keough, Loeb, Perry, and Rombach this completes the conjecture of Alikhani and Soltani.