# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

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# Discussiones Mathematicae Graph Theory

## GRAPH COLORINGS WITH LOCAL CONSTRAINTS   A SURVEY

Zsolt Tuza

Computer and Automation Institute
H-1111 Budapest, Kende u. 13-17, Hungary

e-mail: tuza@lutra.sztaki.hu

## Abstract

We survey the literature on those variants of the chromatic number problem where not only a proper coloring has to be found (i.e., adjacent vertices must not receive the same color) but some further local restrictions are imposed on the color assignment. Mostly, the list colorings and the precoloring extensions are considered.

In one of the most general formulations, a graph G = (V,E), sets L(v) of admissible colors, and natural numbers cv for the vertices v ∈ V are given, and the question is whether there can be chosen a subset C(v) ⊆ L(v) of cardinality cv for each vertex in such a way that the sets C(v),C(v') are disjoint for each pair v,v' of adjacent vertices. The particular case of constant |L(v)| with cv = 1 for all v ∈ V leads to the concept of choice number, a graph parameter showing unexpectedly different behavior compared to the chromatic number, despite these two invariants have nearly the same value for almost all graphs.

To illustrate typical techniques, some of the proofs are sketched.

Keywords: graph coloring, list coloring, choice number, precoloring extension, complexity of algorithms, chromatic number.

1991 Mathematics Subject Classification: Primary 05--02, 05C15, Secondary 68R10

## Contents

 0.  Introduction 163 0.1  Standard Definitions 165 0.2  Notation for Vertex Colorings 165 0.3  Some Variations 167 0.4  Small Uncolorable Graphs 167 1.  General Results 168 1.1  Equivalent Formulations 168 1.2  Complete Bipartite Graphs and the Construction of Hajós 170 1.3  Typical Behavior of the Choice Number 172 1.4  Unions of Graphs and the (am,bm)-Conjecture 173 1.5  Graphs and Their Complements 175 2.  Vertex Degrees 176 2.1  The Theorems of Brooks and Gallai 177 2.2  Lower Bounds on the Choice Number 180 2.3  Graph Polynomials 181 2.4  Orientations and Eulerian Subdigraphs 183 3.  Comparisons of Coloring Parameters 185 3.1  Planar Graphs 186 3.2  Graphs with Equal Chromatic and Choice Number 188 3.3  Edge and Total Colorings 191 3.4  Choice Ratio and Fractional Chromatic Number 197 3.5  The Chromatic Polynomial 198 4.  Algorithmic Complexity 200 4.1  Precoloring Extension 202 4.2  Good Characterizations 204 4.3  List Colorings 206 4.4  Choosability 211 4.5  Graph Coloring Games 213

## Acknowledgements

I am indebted to N. Alon, J. A. Bondy, M. Hujter, T. R. Jensen, H. Kierstead, J. Kratochví l, P. Mihók, C. Thomassen, B. Toft, and M. Voigt for fruitful and inspirative discussions on the subject, to G. Bacsó, K. M. Hangos, and T. R. Jensen for their many valuable comments on the first version of the paper, and to N. Eaton, C. N. Jagger, and A.V. Kostochka for useful short remarks. I thank the authors of results cited here but not published so far, for kindly informing me about their most recent works. Also, support from the Konrad-Zuse-Zentrum für Informationstechnik Berlin is thankfully acknowledged, where part of this research was carried out.