[Paper Review] On uniquely list colorable graphs

M. Ghebleh, E. S. Mahmoodian|ArXiv.org|Jun 2, 1999

graph theory and CDMA systems3 citations

TL;DR

This paper investigates uniquely k-list colorable graphs—graphs with a unique proper list coloring for given k-color lists at each vertex. It extends prior work on uniquely 2-list colorable graphs, establishes connections to defining sets in graph coloring and critical sets in Latin squares, and provides foundational results toward characterizing uniquely k-list colorable graphs for k ≥ 3.

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.

Motivation & Objective

• To extend the characterization of uniquely list colorable graphs beyond the known case of k=2.
• To explore structural properties of uniquely k-list colorable graphs for k ≥ 3.
• To establish connections between uniquely list colorable graphs and combinatorial concepts such as defining sets in graph coloring.
• To draw parallels between uniquely list colorable graphs and critical sets in Latin squares.
• To lay foundational results toward a complete classification of uniquely k-list colorable graphs.

Proposed method

• Analyzes the structure of graphs that admit exactly one proper list coloring under given k-color lists per vertex.
• Applies techniques from graph coloring and combinatorial design theory to derive necessary conditions for unique list colorability.
• Uses known results on uniquely 2-list colorable graphs as a base case for generalization.
• Explores the relationship between list colorings and defining sets, where a minimal set of pre-colored vertices forces a unique coloring.
• Draws analogies with critical sets in Latin squares, where a minimal set of entries uniquely determines the full square.
• Employs combinatorial arguments and structural graph theory to identify constraints on the existence and form of uniquely k-list colorable graphs.

Experimental results

Research questions

• RQ1 What structural properties must a graph possess to be uniquely k-list colorable for k ≥ 3?
• RQ2 How do uniquely k-list colorable graphs relate to defining sets in graph coloring?
• RQ3 In what ways do uniquely k-list colorable graphs mirror the behavior of critical sets in Latin squares?
• RQ4 What are the necessary and sufficient conditions for a graph to be uniquely k-list colorable?
• RQ5 How can the characterization of uniquely 2-list colorable graphs be generalized to higher k?

Key findings

• The paper establishes foundational results that pave the way for characterizing uniquely k-list colorable graphs beyond the k=2 case.
• It confirms a deep connection between uniquely list colorable graphs and defining sets in graph colorings.
• The work reveals a structural analogy between uniquely list colorable graphs and critical sets in Latin squares.
• The authors provide a framework for analyzing higher-order uniquely list colorable graphs using combinatorial and graph-theoretic tools.
• The results are presented as a stepping stone toward a full classification of such graphs, with implications for list coloring and design theory.
• The journal publication in *Ars Combinatoria* (2001) confirms the paper’s contribution to the combinatorial literature.