A technique for multicoloring triangle-free hexagonal graphs

In order to avoid interference in cellular telephone networks, sets of radio frequencies are to be assigned to transmitters such that adjacent transmitters are allotted disjoint sets of frequencies. Often these transmitters are laid out like vertices of a triangular lattice in a plane. This problem corresponds to the problem of multicoloring an induced subgraph of a triangular lattice with integer demands associated with each vertex. We deal with the simpler case of triangle-free subgraphs of the lattice. [Frédéric Havet, Discrete Math. 233 (2001) 1–3] uses inductive arguments to prove that triangle-free hexagonal graphs can be colored with image colors where image is the maximum demand on a clique in the graph. We give a simpler proof and hope that our techniques can be used to prove the conjecture by [McDiarmid and Reed, Networks Suppl. 36 (2000) 114–117] that these graphs are image-multicolorable.