Acyclic colourings of graphs with bounded degree

TL;DR

This paper investigates acyclic colourings of graphs with bounded degree, establishing new bounds for such colourings, proving NP-completeness for certain cases, and providing a linear-time algorithm for large colour sets.

Contribution

It introduces new bounds for acyclic colourings with degree constraints, proves NP-completeness for specific acyclic 2-colouring problems, and offers an efficient algorithm for acyclic t-improper colourings.

Findings

Any graph with maximum degree 5 has an acyclic 5-colouring with each class having max degree 4.

Deciding acyclic 2-colouring with max degree 3 per class is NP-complete for graphs with degree 5.

A linear-time algorithm exists for acyclic t-improper colouring with sufficiently many colours.

Abstract

A $k$-colouring (not necessarily proper) of vertices of a graph is called {\it acyclic}, if for every pair of distinct colours $i$ and $j$ the subgraph induced by the edges whose endpoints have colours $i$ and $j$ is acyclic. In the paper we consider some generalised acyclic $k$-colourings, namely, we require that each colour class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic $5$-colouring such that each colour class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph $G$ has an acyclic 2-colouring in which each colour class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic $t$-improper colouring of any graph with maximum degree $d$ assuming that the number of colors is large enough.