Colourability and word-representability of near-triangulations [article]

Marc Elliot Glen
2018 arXiv   pre-print
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
more » ... ntable 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].
arXiv:1605.01688v4 fatcat:xarieqfk2ngrrnvxvgdyhws6xq