On the complexity of colouring antiprismatic graphs

TL;DR

This paper investigates the computational complexity of coloring antiprismatic graphs, providing polynomial algorithms for clique cover and vertex-disjoint triangles problems in specific subclasses, advancing understanding of their structural properties.

Contribution

It introduces polynomial-time algorithms for clique cover in non-orientable prismatic graphs and for vertex-disjoint triangles in all prismatic graphs, leveraging structural insights.

Findings

Polynomial algorithm for clique cover in non-orientable prismatic graphs.

Polynomial algorithm for vertex-disjoint triangles in all prismatic graphs.

Structural description aids in algorithm development for these classes.

Abstract

A graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called \emph{antiprismatic}. The complexity of colouring antiprismatic graphs is still unknown. Equivalently, the complexity of the clique cover problem in prismatic graphs is not known. Chudnovsky and Seymour gave a full structural description of prismatic graphs. They showed that the class can be divided into two subclasses: the orientable prismatic graphs, and the non-orientable prismatic graphs. We give a polynomial time algorithm that solves the clique cover problem in every non-orientable prismatic graph. It relies on the the structural description and on later work of Javadi and Hajebi. We give a polynomial time algorithm which solves the vertex-disjoint triangles problem for every prismatic graph. It does not rely on the structural description.