Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal b-Colourings and Fall Colourings in $H$-Free Graphs

Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, David Manlove, Fabricio Mendoza, Daniël Paulusma

Abstract

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in $H$-free graphs. For Tight b-Chromatic Number in $H$-free graphs, we develop a general technique to determine new graphs $H$, for which the problem is polynomial-time solvable, and we also determine new graphs $H$, for which the problem is still NP-complete. We show, for the first time, the existence of a graph $H$ such that in $H$-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.

Optimal b-Colourings and Fall Colourings in $H$-Free Graphs

Abstract

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in -free graphs. For Tight b-Chromatic Number in -free graphs, we develop a general technique to determine new graphs , for which the problem is polynomial-time solvable, and we also determine new graphs , for which the problem is still NP-complete. We show, for the first time, the existence of a graph such that in -free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.

Paper Structure

This paper contains 8 sections, 25 theorems, 1 equation, 6 figures.

Table of Contents

  1. Introduction
  2. Known Complexity Results
  3. Our Results
  4. Preliminaries
  5. The Proof of Theorem \ref{['t-b']}
  6. The Proof of Theorem \ref{['t-tight']}
  7. The Proof of Theorem \ref{['t-fall']}
  8. Conclusions

Key Result

Theorem 3

For a graph $H$, b-Chromatic Number in $H$-free graphs is polynomial-time solvable if $H\subseteq_i P_4$, and NP-hard otherwise.

Figures (6)

  • Figure 1: An example of a b-colouring of a $(2n-1)$-vertex complete bipartite graph minus a matching of size $n-1$ (left) with $n$ colours that is not a fall colouring, and an example of a fall colouring of a $2n$-vertex complete bipartite graph minus a perfect matching (right) with $n$ colours.
  • Figure 2: Gadget $H_{uv}$ used in the NP-hardness proof of b-Chromatic Number on co-bipartite graphs BSSV15.
  • Figure 3: On the left: a tight graph $G$ with a set $T$ of dense vertices, where $G$ is annotated with an $S'$-partial b-colouring $c'$ (as per Definition \ref{['d-precoldefs']}) using colours $\{1,\ldots,5\}$. On the right: a b-precolouring extension of $c'$ (as per Definition \ref{['d-extension']}).
  • Figure 6: Illustration of the graph $H$ from HSS12.
  • Figure 9: The $C_3$-free gadget $G'$ with $\mathcal{F}(G')=\{3\}$ from the proof of Lemma \ref{['l-fallbip']}.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 6: Ol88
  • Corollary 7: Ol88
  • Lemma 8: BSSV15
  • Lemma 9: KTV02Ma98
  • Lemma 10: HSS12
  • Lemma 11: BDMMV09
  • Theorem 11
  • ...and 17 more