The Distinguishing Index of Mycielskian Graphs

TL;DR

This paper investigates the symmetry measure called the distinguishing index in Mycielskian graphs, proving bounds that confirm and extend a 2020 conjecture, thereby advancing understanding of graph automorphisms.

Contribution

The authors prove that the distinguishing index of the Mycielskian of certain graphs does not exceed that of the original graph, confirming and extending a conjecture from 2020.

Findings

Proved $ ext{Dist'}( ext{μ}(G)) \

Confirmed the conjecture for a broad class of graphs, including generalized Mycielskian graphs.

Extended the conjecture's validity, completing its proof with recent related work.

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.