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Art galleries as described in the article (as triangulations of polygons with some interior lines missing, the interior lines being added as part of the proof) are outerplanar. An earlier proof that the chromatic number of an outerplanar graph is at most three  and that a triangulation of a polygon not only has chromatic number three but that there is only one way (except for interchange of color names) to achieve such a coloring  was given by Chartrand and Geller, and appears as an exercise in Frank Harary's textbook (page 149).