On the -labelings of amalgamations of graphs

Abstract: The problem of assigning frequencies to transmitters in a radio network can be modeled through vertex labelings of a graph, wherein each vertex represents a transmitter and edges connect vertices whose corresponding transmitters are operating in close proximity. In one such model, an -labeling of a graph G is employed, which is an assignment fof nonnegative integers to the vertices of G such that if vertices x and y are adjacent, , and if x and y are at distance two, . The -number of G is the minimum span over all -labelings of G. Informally, an amalgamation of two graphs and along a fixed graph is the simple graph obtained by identifying the vertices of two induced subgraphs isomorphic to , one of and the other of . We provide upper bounds for the -number of the amalgamation of graphs along a given graph by determining the exact -number of amalgamations of complete graphs along a complete graph. We also provide the exact -numbers of amalgamations of rectangular grids along a path, or more specifically, of the Cartesian products of a path and a star with spokes of arbitrary lengths. [Copyright &y& Elsevier]