Monochromatic Paths and Triangulated Graphs

This paper considers two properties of graphs, one geometrical and one topolog-ical, and shows that they are strongly related. Let G be a graph with four distinguished and distinct vertices, w 1 ; w 2 ; b 1 ; b 2. Consider the two properties, T RI + (G) and M ON O(G), deened as follows. T RI + (G): There is a planar drawing of G such that: all 3-cycles of G are faces; all faces of G are triangles except for the single face which is the 4-cycle (w 1 ? b 1 ? w 2 ? b 2 ? w 1). M ON O(G): G contains the 4-cycle (w 1 ?b 1 ?w 2 ?b 2 ?w 1), and for any labeling of the vertices of G by the colors fwhite, blackg, such that w 1 and w 2 are white, while b 1 and b 2 are black, precisely one of the following holds. There is a path of white vertices connecting w 1 and w 2. There is a path of black vertices connecting b 1 and b 2. Our main result is that a graph G enjoys property T RI + (G) if, and only if, it is minimal with respect to property M ON O. Building on this, we show that one can decide in polynomial time whether or not a given graph G has property M ON O(G).