Labelings in Cayley digraphs
Graph labeling, introduced by \textit{A. Rosa} in [``On certain valuations of the vertices of a graph,'' Theory Graphs, Int. Symp. Rome 1966, 349--355 (1967; Zbl 0193.53204)], deals with assigning integers to the vertices or edges or both satisfying some combinatorial constraint. Labeled graphs serve as useful models for numerous applications such as a circuit design, radar, communication network, etc. There are different kinds of labelings, but to study labeling of Cayley graphs a new kind of labeling called super-vertex \((a,d)\)-antimagic labeling for digraphs was introduced. In this paper, the authors show the existence of supervertex \((a,d)\)-antimagic labeling and vertex-magic total labeling for some Cayley digraphs. The following are the main theorems: Theorem 1: The Cayley digraph associated with dihedral group \(D_n\) admits super-vertex \((a,d)\)-antimagic labeling. Theorem 2: The Cayley digraph associated with the group \(Z_n\) with the generating set \((a,b,b+k)\) admits magic labeling. Illustrative examples to support both results above have been provided for \(n=8\), and \(12\). An algorithm for obtaining super-vertex \((a,d)\)-antimagic labeling for the Cayley digraphs associated with \(D_n\), and also an algorithm for getting magic labeling for the Cayley digraphs associated with the \(Z_n\) group.
scientific article; zbMATH DE number 5817924
scientific article