Tilings with trichromatic colored-edges triangles

This paper studies the tilings with colored-edges triangles constructed on a triangulation of a simply connected orientable surface such that the degree of each interior vertex is even (such as, for (fundamental) example, a part of the triangular lattice of the plane). The constraints are that we only use three colors, all the colors appear in each tile and two tile can share an edge only if this edge has the same color in both tiles. Using previous results on lozenge tilings, we give a linear algorithm of coloration for triangulations of the sphere, or of planar regions with the constraint that the boundary is monochromatic. We de1ne a 2ip as a shift of colors on a cycle of edges using only two colors. We prove 2ip connectivity of the set of solutions for the cases seen above (i.e. two tiling are mutually accessible by a sequence of 2ips), and prove that there is no 2ip accessibility in the general case where the boundary is not assumed to be monochromatic. Nevertheless, using 2ips, we obtain a tiling invariant, even in the general case. We 1nish relaxing the condition, allowing monochromatic triangles. With this hypothesis, some local 2ips are su5cient for connectivity. We give a linear algorithm of coloration, and strong structural results on the set of solutions. c © 2004 Elsevier B.V. All rights reserved.