A structural description of Zykov and Blanche Descartes graphs

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.

Figures (9)

• Figure 1: Graphs $F$ and $F'$
• Figure 2: Graph $H$
• Figure 3: Non-Zykov graph of treewidth $2$
• Figure 4: Graph $G_i$
• Figure 5: Graph $G_{C_j}$

Theorems & Definitions (50)

• lemma thmcounterlemma
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