Coloring bridge-free antiprismatic graphs

TL;DR

This paper presents a polynomial-time algorithm for the clique cover problem in co-bridge-free prismatic graphs, a class of graphs for which the coloring problem's complexity was previously unresolved.

Contribution

It introduces a structural characterization of co-bridge-free prismatic graphs and leverages this to solve the clique cover problem efficiently.

Findings

Polynomial-time algorithm for clique cover in co-bridge-free prismatic graphs

Bounded number of disjoint triangles in these graphs

Application of existing structural theorems to solve the problem

Abstract

The coloring problem is a well-research topic and its complexity is known for several classes of graphs. However, the question of its complexity remains open for the class of antiprismatic graphs, which are the complement of prismatic graphs and one of the four remaining cases highlighted by Lozin and Malishev. In this article we focus on the equivalent question of the complexity of the clique cover problem in prismatic graphs. A graph $G$ is prismatic if for every triangle $T$ of $G$, every vertex of $G$ not in $T$ has a unique neighbor in $T$. A graph is co-bridge-free if it has no $C_4+2K_1$ as induced subgraph. We give a polynomial time algorithm that solves the clique cover problem in co-bridge-free prismatic graphs. It relies on the structural description given by Chudnovsky and Seymour, and on later work of Preissmann, Robin and Trotignon. We show that co-bridge-free prismatic graphs have a bounded number of disjoint triangles and that implies that the algorithm presented by Preissmann et al. applies.