Roman domination is an historically inspired variety of domination in graphs, in which vertices are assigned a value from the set $\{0,1,2\}$ in such a way that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. The Roman domination number is the minimum possible sum of all values in such an assignment. Using an algebraic approach we  present an $O(C)$-time algorithm for computing the Roman domination numbers of special classes of graphs called polygraphs, which include rotagraphs and fasciagraphs. Using this algorithm we determine formulas for the Roman domination numbers of the Cartesian products of the form $P_n\Box P_k$, $P_n\Box C_k$, for $k\leq8$ and $n \in {\mathbb N}$, and $C_n\Box P_k$ and $C_n\Box C_k$, for $k\leq 6$ and $n \in {\mathbb N}$, for paths $P_n$ and cycles $C_n$. We also find all special graphs called Roman graphs in these families of graphs.

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An 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. Frequency assignment problem can be abstracted as a multicoloring problem on a weighted hexagonal graph, where the weights represent the number of calls to be assigned at vertices. In this paper we present a distributed algorithm for multicoloring hexagonal graphs using only the local clique numbers ω 1 (v) and ω 2 (v) at each vertex v of the given hexagonal graph, which can be computed from local information available at the vertex. We prove the algorithm uses no more than 4ω(G)/3 colors for any hexagonal graph G, without explicitly computing the global clique number ω(G). We also prove that our algorithm is 2-local, i.e., the computation at a vertex v ∈ G uses only information about the demands of vertices whose graph distance from v is less than or equal to 2.

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In a fundamental paper, G. Sabidussi [“Graph Multiplication,” Mathematische Zeitschrift, Vol. 72 (1960), pp. 446–457] used a tower of equivalence relations on the edge set E(G) of a connected graph G to decompose G into a Cartesian product of prime graphs. Later, a method by R.L. Graham and P.M. Winkler [“On Isometric Embeddings of Graphs,” Transactions of the American Mathematics Society, Vol. 288 (1985), pp. 527–533] of embedding a connected graph isometrically into Cartesian products opened another approach to this problem. In both approaches an equivalence relation σ that determines the prime factorization is constructed. The methods differ by the starting relations used. We show that σ can be obtained as the convex hull of the starting relation used by Sabidussi. Our result also holds for the relation determining the prime decomposition of infinite connected graphs with respect to the weak Cartesian product. Moreover, we show that this relation is the transitive closure of the union of the starting relations of Sabidussi and Winkler [“Factoring a Graph in Polynomial Time,” European Journal of Combinatorics, Vol. 8 (1987), pp. 209–212], thereby generalizing the result of T. Feder [“Product Graph Representations,” Journal of Graph Theory, Vol 16 (1993), pp. 467–488] from finite to infinite graphs.

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