3-Colourability of Dually Chordal Graphs in Linear Time

TL;DR

This paper proves that 3-colourability of dually chordal graphs can be decided in linear time, while 4-colourability remains NP-complete, providing a complete characterization for 3-colourability.

Contribution

It establishes a linear-time algorithm for 3-colourability of dually chordal graphs and characterizes when such graphs are 3-colourable.

Findings

3-colourability is solvable in linear time for dually chordal graphs.

4-colourability is NP-complete for dually chordal graphs.

A dually chordal graph is 3-colourable iff it is perfect and has no clique of size four.

Abstract

A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four.