Coloring bridge-free antiprismatic graphs
Cléophée Robin, Eileen Robinson
TL;DR
This work resolves the complexity of vertex coloring for bridge-free antiprismatic graphs by translating the problem to clique cover in co-bridge-free prismatic graphs. It leverages a detailed structural description (prime graphs, worn chain decomposition, and Schlaefli-prismatic graphs) to show that every such graph either has a triangle hitting set of size at most 5 or is a Schlaefli-prismatic exception. Building on this dichotomy and the matrix-based framework of Preissmann et al., the authors derive a polynomial-time clique-cover algorithm with runtime $O(n^{7.5})$, which yields a polynomial vertex coloring algorithm for bridge-free antiprismatic graphs. The results provide a near-complete complexity classification for this graph family and pave the way for extensions to additional forbidden-subgraph configurations and parameterized approaches around the hitting-set size.
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.