Acyclic, Star, and Injective Colouring: Bounding the Diameter

TL;DR

This paper investigates how bounding the diameter of graphs affects the complexity of acyclic, star, and injective colourings, revealing new polynomial and NP-complete cases and a complexity jump for star 3-colouring.

Contribution

It provides almost-complete complexity classifications for these colouring problems on graphs with bounded diameter, including the first known complexity jump for star 3-colouring.

Findings

Acyclic 3-Colouring is polynomial-time for diameter ≤ 2 but NP-complete for diameter ≥ 4.

Star 3-Colouring is polynomial-time for diameter ≤ 3 but NP-complete for diameter ≥ 8.

L(1,2)-Labelling is NP-complete for graphs of diameter 2.

Abstract

We examine the effect of bounding the diameter for well-studied variants of the Colouring problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring. The last problem is also known as $L(1,1)$-Labelling and we also consider the framework of $L(a,b)$-Labelling. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most $d$, Acyclic $3$-Colouring is polynomial-time solvable if $d\leq 2$ but NP-complete if $d\geq 4$, and Star $3$-Colouring is polynomial-time solvable if $d\leq 3$ but NP-complete for $d\geq 8$. As far as we are aware, Star $3$-Colouring is the first problem that exhibits a complexity jump for some $d\geq 3$. Our third main result is that $L(1,2)$-Labelling is NP-complete for graphs of diameter $2$; we relate the latter problem to a special case of Hamiltonian Path.