Finding a five bicolouring of a triangle-free subgraph of the triangular lattice

A basic problem in the design of mobile telephone networks is to assign sets of radio frequency bands (colours) to transmitters (vertices) to avoid interference. Often the transmitters are laid out like vertices of a triangular lattice in the plane. We investigate the corresponding colouring problem of assigning sets of colours of size p(v) to each vertex of the triangular lattice so that the sets of colours assigned to adjacent vertices are disjoint. A n-[p℄colouring of a graph G is a mapping from V (G) into the set of the subsets of f1; 2; : : : ; ng such that j (v)j = p(v) and for any adjacent vertices u and v, (u)\ (v) = ;. We give here an alternative proof of the fact that every triangular-free induced subgraph of the triangular lattice is 5-[2]colourable. This proof yields a constant time distributed algorithm that finds a 5-[2]colouring of such a graph. We then give a distributed algorithm that finds a [p℄colouring of a triangle-free induced subgraph of the triangular lattice with at most 5!p(G) 4 + 3 colours.