On the L(2,1)-labelings of amalgamations of graphs

  • Authors:

  • Sarah Spence Adams;Noura Howell;Nathaniel Karst;Denise Sakai Troxell;Junjie Zhu

  • Affiliations:

  • Franklin W. Olin College of Engineering, Olin Way, Needham, MA 02492, USA;Franklin W. Olin College of Engineering, Olin Way, Needham, MA 02492, USA;Mathematics and Sciences Division, Babson College, Babson Park, MA 02457, USA;Mathematics and Sciences Division, Babson College, Babson Park, MA 02457, USA;Franklin W. Olin College of Engineering, Olin Way, Needham, MA 02492, USA

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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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 L(2,1)-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, |f(x)-f(y)|=2, and if x and y are at distance two, |f(x)-f(y)|=1. The @l-number of G is the minimum span over all L(2,1)-labelings of G. Informally, an amalgamation of two graphs G"1 and G"2 along a fixed graph G"0 is the simple graph obtained by identifying the vertices of two induced subgraphs isomorphic to G"0, one of G"1 and the other of G"2. We provide upper bounds for the @l-number of the amalgamation of graphs along a given graph by determining the exact @l-number of amalgamations of complete graphs along a complete graph. We also provide the exact @l-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.