Monochromatic Paths and Triangulated Graphs

This paper considers two properties of graphs, one geometrical and one topological, 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, TRI + (G) and MONO(G), defined as follows. TRI + (G): There is a planar drawing of G such that: ffl all 3-cycles of G are faces; ffl all faces of G are triangles except for the single face which is the 4-cycle (w 1 \Gamma b 1 \Gamma w 2 \Gamma b 2 \Gamma w 1 ). MONO(G): G contains the 4-cycle (w 1 \Gamma b 1 \Gamma w 2 \Gamma b 2 \Gamma 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. ffl There is a path of white vertices connecting w 1 and w 2 . ffl There is a path of black vertices connecting b 1 and b 2 . Our main result is that a graph G enjoys property TRI + (G) if, and only if, it is minimal w..