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Graph Colouring: A Visual Tour

Rhyd Lewis

TL;DR

The topics of node, edge, and face colouring along with their associated algorithms are considered along with their associated algorithms to enhance understanding and appeal.

Abstract

Graph colouring is a combinatorial optimisation problem with applications in several important domains, including sports scheduling, cartography, street map navigation, and timetabling. It is also of significant theoretical interest and a standard subject in university-level courses on graph theory, algorithms, and combinatorics. In this paper, we consider the topics of node, edge, and face colouring along with their associated algorithms. Theoretical results are reviewed and brought to life through a collection of detailed, visually engaging figures designed to enhance understanding and appeal.

Graph Colouring: A Visual Tour

TL;DR

The topics of node, edge, and face colouring along with their associated algorithms are considered along with their associated algorithms to enhance understanding and appeal.

Abstract

Graph colouring is a combinatorial optimisation problem with applications in several important domains, including sports scheduling, cartography, street map navigation, and timetabling. It is also of significant theoretical interest and a standard subject in university-level courses on graph theory, algorithms, and combinatorics. In this paper, we consider the topics of node, edge, and face colouring along with their associated algorithms. Theoretical results are reviewed and brought to life through a collection of detailed, visually engaging figures designed to enhance understanding and appeal.
Paper Structure (7 sections, 1 theorem, 14 figures, 1 algorithm)

This paper contains 7 sections, 1 theorem, 14 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Node Colouring
  3. Algorithms for Node Colouring
  4. Visualising Node Colourings
  5. Edge Colouring
  6. Face Colouring
  7. Conclusions

Key Result

Theorem 1

A connected planar graph $G$ is Eulerian if and only if its dual graph $G^*$ is bipartite.

Figures (14)

  • Figure 1: A node colouring, edge colouring, and face colouring (respectively) of the dodecahedral graph. This graph has $n=20$ nodes and $m=30$ edges. As shown, its chromatic number and chromatic index are both three. Because it is planar, the faces of this graph can also be coloured. In this case, the face chromatic number is four.
  • Figure 2: Performance of the HEA and backtracking algorithms using $G(n,p)$ graphs for $p=0.1$, $0.5$, and $0.9$ respectively. Each point and box plot, generated by the backtracking algorithm and HEA, respectively, was generated from runs on fifty randomly generated graphs.
  • Figure 3: Three ways of drawing a node colouring of a $G(30,0.5)$ graph. In this case $\chi(G)=7$. Despite looking different, the graph and colouring is identical in each image.
  • Figure 4: Optimal node colourings of bipartite graphs. The top three images show binary trees with $64$, $256$, and $512$ nodes; the bottom three show a square lattice, a hexagonal lattice, and the great rhombicosidodecahedral graph (HoG 1122). In all cases $\chi(G)=2$.
  • Figure 5: Optimal node colourings of a triangular lattice graph, Thomassen graph (HoG 1347), and the great rhombicosidodecahedral line graph (HoG 51392). In all cases $\chi(G)=3$.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: DSatur node selection rule
  • Theorem 1