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Results and Problems on Equitable Coloring of Graphs

H. A. Kierstead, Alexandr Kostochka, Zimu Xiang

TL;DR

This paper surveys equitable coloring, focusing on bounds via maximum degree and Ore-degree, equitable list coloring, and several variants. It provides a self-contained proof and polynomial-time algorithm for equitable $k$-coloring when $\Delta(G)<k$, and extends the theory to directed graphs and Ore-degree constraints, contributing new existence results and algorithmic techniques. It also discusses the Chen–Lih–Wu conjecture and its Ore-version, along with extensive results in equitable list coloring, SE list coloring, and sparse-graph regimes, highlighting open questions and conjectures. The work has algorithmic implications for scheduling and resource allocation, and deepens understanding of how structural graph properties govern the feasibility of balanced colorings in both standard and list settings.

Abstract

A proper coloring of vertices of a graph is equitable if the sizes of any two color classes differ by at most 1. Such colorings have many applications and are interesting by themselves. In this paper, we discuss the state of art and unsolved problems on equitable coloring and its list versions.

Results and Problems on Equitable Coloring of Graphs

TL;DR

This paper surveys equitable coloring, focusing on bounds via maximum degree and Ore-degree, equitable list coloring, and several variants. It provides a self-contained proof and polynomial-time algorithm for equitable -coloring when , and extends the theory to directed graphs and Ore-degree constraints, contributing new existence results and algorithmic techniques. It also discusses the Chen–Lih–Wu conjecture and its Ore-version, along with extensive results in equitable list coloring, SE list coloring, and sparse-graph regimes, highlighting open questions and conjectures. The work has algorithmic implications for scheduling and resource allocation, and deepens understanding of how structural graph properties govern the feasibility of balanced colorings in both standard and list settings.

Abstract

A proper coloring of vertices of a graph is equitable if the sizes of any two color classes differ by at most 1. Such colorings have many applications and are interesting by themselves. In this paper, we discuss the state of art and unsolved problems on equitable coloring and its list versions.

Paper Structure

This paper contains 16 sections, 34 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Framing the theory
  3. Organization
  4. Notation
  5. Theorems related to bounded maximal degree
  6. Setup for equitable coloring in terms of maximum degree
  7. Finishing the proof
  8. A polynomial-time algorithm
  9. Equitable coloring in terms of maximum Ore-degree
  10. Equitable coloring of directed graphs
  11. Chen-Lih-Wu Conjecture
  12. An equivalent conjecture
  13. An Ore version of the Chen-Lih-Wu Conjecture
  14. Equitable list coloring
  15. Coloring sparse graphs with fewer colors
  16. ...and 1 more sections

Key Result

Theorem 1

A graph $G$ on $n$ vertices with maximum degree $\Delta$ is equitably k-colorable if $\Delta<k$ and either

Figures (3)

  • Figure 3.1: One $5$-equitable and three $4$-equitable basic graphs.
  • Figure 3.2: Six $3$-equitable basic graphs.
  • Figure 5.1: Graph on the left of the picture has $27$ vertices, $32$ edges and cannot be equitably $3$-colored. Graph on the right of the picture has $13$ vertices, $17$ edges and cannot be equitably $4$-colored.

Theorems & Definitions (60)

  • Conjecture : Erdős E, 1964
  • Conjecture : Grünbaum Grun, 1968
  • Theorem 1
  • Theorem 2: Hajnal and Szemerédi HSz, 1970
  • Theorem 3: Bollobás and Guy BG83, 1983
  • proof : Proof of thm:H-Sz
  • Theorem 4: Kierstead, Kostochka, Mydlarz and SzemerédiMR2676836, 2010
  • proof
  • Lemma 5
  • proof
  • ...and 50 more