5 research outputs found
Radio Graceful Labelling of Graphs
Radio labelling problem of graphs have their roots in communication problem known as \emph{Channel Assignment Problem}. For a simple connected graph , a radio labeling is a mapping such that for each pair of distinct vertices , where is the diameter of and is the distance between and . A radio labeling of a graph is a \emph{radio graceful labeling} of if . A graph for which a radio graceful labeling exists is called \emph{radio graceful}. In this article, we study radio graceful labeling for general graphs in terms of some new parameters
Radio Number of Hamming Graphs of Diameter 3
For a simple, connected graph, a vertex labeling is called a \emph{radio labeling of } if it satisfies |f(u)-f(v)|\geq\diam(G)+1-d(u,v) for all distinct vertices . The \emph{radio number of } is the minimal span over all radio labelings of . If a bijective radio labeling onto exists, is called a \emph{radio graceful} graph. We determine the radio number of all diameter 3 Hamming graphs and show that an infinite subset of them is radio graceful
Radio Graceful Hamming Graphs
For and a simple, connected graph, a -radio labeling of requires all pairs of distinct vertices and to satisfy . We consider -radio labelings of when . In this setting, is injective; if is also surjective onto , then is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio graceful Hamming graphs. The main result shows that the Cartesian product of copies of a complete graph is radio graceful for certain . Graphs of this form provide infinitely many examples of radio graceful graphs of arbitrary diameter. We also show that these graphs are not radio graceful for large
Radio Graceful Hamming Graphs
For k ∈ ℤ+ and G a simple, connected graph, a k-radio labeling f : V (G) → ℤ+ of G requires all pairs of distinct vertices u and v to satisfy |f(u) − f(v)| ≥ k + 1 − d(u, v). We consider k-radio labelings of G when k = diam(G). In this setting, f is injective; if f is also surjective onto {1, 2, . . . , |V (G)|}, then f is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio graceful Hamming graphs. The main result shows that the Cartesian product of t copies of a complete graph is radio graceful for certain t. Graphs of this form provide infinitely many examples of radio graceful graphs of arbitrary diameter. We also show that these graphs are not radio graceful for large t