On uniquely list colorable graphs

TL;DR

This paper explores the properties of uniquely list colorable graphs, extending previous characterizations for the case of k=2 and establishing connections to graph coloring sets and Latin squares.

Contribution

It provides new results towards characterizing uniquely k-list colorable graphs for general k, building on prior work for k=2, and links this concept to other combinatorial structures.

Findings

Characterization results for uniquely k-list colorable graphs

Connections between list coloring and Latin squares

Foundational results for future graph coloring research

Abstract

Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2-list colorable graphs. Here we state some results which will pave the way in characterization of uniquely k-list colorable graphs. There is a relationship between this concept and defining sets in graph colorings and critical sets in latin squares.