The List Distinguishing Number Equals the Distinguishing Number for Interval Graphs

TL;DR

This paper proves that for interval graphs, the minimum number of colors needed for a distinguishing coloring is the same whether colors are chosen freely or from lists, establishing an equality of these two parameters.

Contribution

The paper establishes that the distinguishing number and list distinguishing number are equal for all interval graphs, a significant result in graph symmetry and coloring theory.

Findings

Distinguishing number equals list distinguishing number for interval graphs

Provides a new insight into symmetry-breaking colorings in interval graphs

Enhances understanding of coloring constraints in graph automorphisms

Abstract

A \textit{distinguishing coloring} of a graph $G$ is a coloring of the vertices so that every nontrivial automorphism of $G$ maps some vertex to a vertex with a different color. The \textit{distinguishing number} of $G$ is the minimum $k$ such that $G$ has a distinguishing coloring where each vertex is assigned a color from $\{1,\ldots,k\}$. A \textit{list assignment} to $G$ is an assignment $L=\{L(v)\}_{v\in V(G)}$ of lists of colors to the vertices of $G$. A \textit{distinguishing $L$-coloring} of $G$ is a distinguishing coloring of $G$ where the color of each vertex $v$ comes from $L(v)$. The {\it list distinguishing number} of $G$ is the minimum $k$ such that every list assignment to $G$ in which $|L(v)|=k$ for all $v\in V(G)$ yields a distinguishing $L$-coloring of $G$. We prove that if $G$ is an interval graph, then its distinguishing number and list distinguishing number are equal.