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

TL;DR

This paper classifies the computational complexity of four graph colouring problems in $H$-free graphs, revealing new polynomial and NP-complete cases and establishing a unique complexity divergence for certain graphs.

Contribution

It provides a complete complexity classification for b-Chromatic, Fall Chromatic, and Fall Achromatic Numbers in $H$-free graphs, introducing new techniques and results.

Findings

Classified complexity of b-Chromatic, Fall Chromatic, and Fall Achromatic Numbers.

Identified new graphs $H$ where problems are polynomial-time solvable or NP-complete.

Demonstrated the first instance where b-Chromatic Number is NP-hard but Tight b-Chromatic Number is polynomial.

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.