A structural description of Zykov and Blanche Descartes graphs

TL;DR

This paper provides a structural characterization of Zykov and Blanche Descartes graphs, revealing their properties, recognition complexity, and fixed-parameter tractability related to treewidth.

Contribution

It introduces a structural description based on splitting stable sets and analyzes recognition complexity and fixed-parameter tractability for these graph classes.

Findings

Recognition of Zykov graphs is NP-complete.

Recognition is fixed-parameter tractable in treewidth.

Structural characterization based on splitting stable sets.

Abstract

In 1949, Zykov proposed the first explicit construction of triangle-free graphs with arbitrarily large chromatic number. We define a Zykov graph as any induced subgraph of a graph created using Zykov's construction. We give a structural characterization of Zykov graphs based on a specific type of stable set, that we call splitting stable set. It implies that recognizing this class is NP-complete, while being FPT in the treewidth of the input graph. We provide similar results for the Blanche Descartes construction.