The chromatic distinguishing index of certain graphs

TL;DR

This paper investigates the minimum number of labels needed for proper edge labelings of graphs that break all non-trivial automorphisms, focusing on specific graphs and graph operations like corona product and join.

Contribution

It computes the distinguishing chromatic index for certain graphs and explores its behavior under corona product and join operations.

Findings

Determined the distinguishing chromatic index for specific graphs.

Analyzed the effect of corona product and join on the index.

Provided formulas or bounds for these graph operations.

Abstract

The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. The distinguishing chromatic index $\chi'_D (G)$ of a graph $G$ is the least number $d$ such that $G$ has a proper edge labeling with $d$ labels that is preserved only by the identity automorphism of $G$. In this paper we compute the distinguishing chromatic index for some specific graphs. Also we study the distinguishing chromatic index of corona product and join of two graphs.