Search

Abstract: An -labeling of a graph is a function such that if and are adjacent vertices, and if and are at distance 2. Such labelings were introduced as a way of modeling the assignment of frequencies to transmitters operating in close proximity within a communications network. The lambda number of is the minimum over all -labelings of . This paper considers the lambda number of the matched sum of two same-order disjoint graphs, wherein the graphs have been connected by a perfect matching between the two vertex sets. Matched sums have been studied in this context to model possible connections between two different networks with the same number of transmitters. We completely determine the lambda number of matched sums where one of the graphs is a complete graph or a complete graph minus an edge. We conclude by discussing some difficulties that are encountered when trying to generalize this problem by removing more edges from a complete graph. [Copyright &y& Elsevier]

Abstract: An -labeling of a graph is an assignment of nonnegative integers to its vertices so that adjacent vertices get labels at least two apart and vertices at distance two get distinct labels. The -number of a graph G, denoted by , is the minimum range of labels taken over all of its -labelings. We show that the -number of the Cartesian product of any two cycles is 6, 7 or 8. In addition, we provide complete characterizations for the products of two cycles with -number exactly equal to each one of these values. [Copyright &y& Elsevier]

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]

Abstract: An -labeling of a graph is an assignment of nonnegative integers to the vertices of such that if vertices and are adjacent, , and if and are at distance two, . The -number of is the minimum span over all -labelings of . A generalized Petersen graph (GPG) of order consists of two disjoint copies of cycles on vertices together with a perfect matching between the two vertex sets. By presenting and applying a novel algorithm for identifying GPG-specific isomorphisms, this paper provides exact values for the -numbers of all GPGs of orders 9, 10, 11, and 12. For all but three GPGs of these orders, the -numbers are 5 or 6, improving the recently obtained upper bound of 7 for GPGs of orders 9, 10, 11, and 12. We also provide the -numbers of several infinite subclasses of GPGs that have useful representations on Möbius strips. [Copyright &y& Elsevier]