Antimagic Labeling of Digraphs

A directed graph D can be labeled antimagically by assigning different integers to its arcs, ensuring that the computed vertex weights are distinct. Antimagic labeling exists in a digraph D with <img src=image/13440650_01.gif> arcs and <img src=image/13440650_02.gif> vertices if it is possible to uniquely match each arc to an integer from 1 to h. For every vertex, the difference between the totals of the labels from incoming arcs and outgoing arcs is unique for each vertex. A directed graph that allows such antimagic labeling is referred to as an antimagic digraph. There are countless methods to create an antimagic digraph. These constructions are essential in fields such as network theory, coding theory, and combinatorial optimization. The subset sum problem is a well-recognized issue in both computer science and combinatorics. These problems play a decisive role in graph labeling, especially when it comes to the creation and evaluation of specific types of labels. In this paper, we connect the idea of subset sum problems with wheel digraphs <img src=image/13440650_03.gif> to represent it as antimagic. The Cartesian product of directed graphs is a fundamental concept in graph theory that holds both theoretical and practical importance. The applications of Cartesian products include network design and analysis, parallel computing, graph decomposition and construction, Game Theory, and Decision-Making. Additionally, in this paper, we have developed antimagic digraphs from the Cartesian product of directed path <img src=image/13440650_04.gif> and <img src=image/13440650_05.gif> by traversing the directed path <img src=image/13440650_06.gif> in alternating and unidirectional ways.