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arXiv · 1405.3527

On word-representability of polyomino triangulations

Prosper Akrobotu·Sergey Kitaev·Zuzana Masárová
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Abstract

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)$ is an edge in $E$. Some graphs are word-representable, others are not. It is known that a graph is word-representable if and only if it accepts a so-called semi-transitive orientation. The main result of this paper is showing that a triangulation of any convex polyomino is word-representable if and only if it is 3-colorable. We demonstrate that this statement is not true for an arbitrary polyomino. We also show that the graph obtained by replacing each $4$-cycle in a polyomino by the complete graph $K_4$ is word-representable. We employ semi-transitive orientations to obtain our results.

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Prosper Akrobotu, Sergey Kitaev, Zuzana Masárová. 2014-05-14. On word-representability of polyomino triangulations. https://arxiv.org/abs/1405.3527

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