The List Distinguishing Number of Kneser Graphs

TL;DR

This paper proves that for Kneser graphs, the list distinguishing number equals the standard distinguishing number, establishing a key equivalence in graph symmetry-breaking colorings.

Contribution

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

Findings

$D_l(G) = D(G)$ for Kneser graphs

List coloring does not increase the distinguishing number

Kneser graphs have equal list and standard distinguishing numbers

Abstract

A graph $G$ is said to be $k$-distinguishable if the vertex set can be colored using $k$ colors such that no non-trivial automorphism fixes every color class, and the distinguishing number $D(G)$ is the least integer $k$ for which $G$ is $k$-distinguishable. If for each $v\in V(G)$ we have a list $L(v)$ of colors, and we stipulate that the color assigned to vertex $v$ comes from its list $L(v)$ then $G$ is said to be $\mathcal{L}$-distinguishable where $\mathcal{L} =\{L(v)\}_{v\in V(G)}$. The list distinguishing number of a graph, denoted $D_l(G)$, is the minimum integer $k$ such that every collection of lists $\mathcal{L}$ with $|L(v)|=k$ admits an $\mathcal{L}$-distinguishing coloring. In this paper, we prove that $D_l(G)=D(G)$ when $G$ is a Kneser graph.