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

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