## AbstractAn important optimization problem in the design of cellular networks is to assign sets of frequencies to transmitters to avoid unacceptable interference. A cellular network is generally modeled as a subgraph of the infinite triangular lattice. The distributed frequency assignment problem can be abstracted as a multicoloring problem on a weighted hexagonal graph, where the weight vector represents the number of calls to be assigned at vertices. In this paper we present a distributed algorithm for 12-[5]coloring of triangle free hexagonal graphs. and a distributed algorithm for multicoloring triangle-free hexagonal graphs with arbitrary demands using only the local clique numbers at each vertex of the given hexagonal graph, which can be computed from local information available at the vertex. We prove that the algorithm uses no more than $\left\lceil \omega(G)/5\right\rceil +C$ colors for any triangle-free hexagonal graph $G$, without explicitly computing the global clique number $\omega(G)$. Hence the competitive ratio of the algorithm is $6/5$.
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