Uniquely Distinguishing Colorable Graphs

TL;DR

This paper explores the unique colorability of graphs with respect to distinguishing colorings, introduces new graph families, and characterizes how the distinguishing chromatic number behaves under graph unions, with applications to disconnected graphs.

Contribution

It introduces two new families of uniquely distinguishing colorable graphs and characterizes the distinguishing chromatic number for various graph unions and classes.

Findings

Every disconnected uniquely distinguishing colorable graph is a union of two isomorphic type 2 graphs.

Any n-colorable uniquely distinguishing tree with n ≥ 3 is a star graph.

Characterizations of graphs based on their distinguishing chromatic number after union operations.

Abstract

A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the distinguishing coloring of a graph and its applications in computing the distinguishing chromatic number of disconnected graphs. We introduce two families of uniquely distinguishing colorable graphs, namely type 1 and type 2, and show that every disconnected uniquely distinguishing colorable graph is the union of two isomorphic graphs of type 2. We obtain some results on bipartite uniquely distinguishing colorable graphs and show that any uniquely distinguishing $n$-colorable tree with $ n \geq 3$ is a star graph. For a connected graph $G$, we prove that $\chi_D(G\cup G)=\chi_D(G)+1$ if and only if $G$ is uniquely distinguishing colorable of type 1. Also, a characterization of all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G) = k$, where $k=n-2, n-1, n$, is given in this paper. Moreover, we determine all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = \ell$, where $\ell=n-1, n, n+1$. Finally, we investigate the family of connected graphs $G$ with $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = 3$.