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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 $xy$ is an edge in $E$. It is known that any word-representable graph $G$ is $k$-word-representable for some $k$, that is, there exists a word $w$ representing $G$ such that each letter occurs exactly $k$ times in $w$. The minimum such $k$ is called $G$'s representation number. A crown graph $H_{n,n}$ is a graph obtained from the complete bipartite graph $K_{n,n}$ by removing a perfect matching. In this paper we show that for $n\geq 5$, $H_{n,n}$'s representation number is $\lceil n/2 \rceil$. This result not only provides a complete solution to the open Problem 7.4.2 in \cite{KL}, but also gives a negative answer to the question raised in Problem 7.2.7 in \cite{KL} on 3-word-representability of bipartite graphs. As a byproduct we obtain a new example of a graph class with a high representation number.

A graph $G = (V,E)$ is word-representable if there is a word $w$ over the alphabet $V$ such that $x$ and $y$ alternate in $w$ if and only if the edge $(x, y)$ is in $G$. It is known [6] that all $3$-colourable graphs are word-representable, while among those with a higher chromatic number some are word-representable while others are not. There has been some recent research on the word-representability of polyomino triangulations. Akrobotu et al.[1] showed that a triangulation of a convex polyomino is word-representable if and only if it is $3$-colourable; and Glen and Kitaev[5] extended this result to the case of a rectangular polyomino triangulation when a single domino tile is allowed. It was shown in [4] that a near-triangulation is $3$-colourable if and only if it is internally even. This paper provides a much shorter and more elegant proof of this fact, and also shows that near-triangulations are in fact a generalization of the polyomino triangulations studied in [1] and [5], and so we generalize the results of these two papers, and solve all open problems stated in [5]., Comment: 7 pages; submitting to PUMA

A recent elegant result of Akrobotu et al. states that a triangulation of any convex polyomino is word-representable if and only if it is 3-colorable. In this paper, we generalize a particular case of this result by showing that the result of Akrobotu et al. is true even if we allow a domino tile, instead of having just $1\times 1$ tiles on a rectangular polyomino., Comment: To appear in the Journal of Combinatorial Mathematics and Combinatorial Computing