  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].
