On Ahn-Hendrey-Kim-Oum question for twin-width of graphs with 6 vertices

TL;DR

This paper proves that all graphs with 6 vertices have twin-width at most 2, resolving an open problem about the maximum twin-width possible for small graphs.

Contribution

It establishes that no 6-vertex graph has twin-width at least 3, extending previous results from graphs with up to 5 vertices.

Findings

All 6-vertex graphs have twin-width ≤ 2

No 6-vertex graph has twin-width ≥ 3

Completes the classification for small graphs' twin-width

Abstract

Twin-width is a recently introduced graph parameter for finite graphs. It is an open problem to determine whether there is an $n$-vertex graph having twin-width at least $n/2$ (due to J. Ahn, K. Hendrey, D. Kim and S. Oum). In an earlier paper, the author showed that such a graph with less than equal to 5 vertices does not exist. In this article, we show that such a graph with 6 vertices does not exist. More precisely, we prove that each graph with 6 vertices has twin-width less than equal to 2.