Acyclic, Star, and Injective Colouring: Bounding the Diameter
Abstract
We examine the effect of bounding the diameter for wellstudied 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 polynomialtime solvable if $d\leq 2$ but NPcomplete if $d\geq 4$, and Star $3$Colouring is polynomialtime solvable if $d\leq 3$ but NPcomplete 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 NPcomplete for graphs of diameter $2$; we relate the latter problem to a special case of Hamiltonian Path.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.10593
 Bibcode:
 2021arXiv210410593B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics