General Position and Edge General Position Problems for Unitary Cayley Graphs: A Direct Product Approach

Let G be a connected graph. A subset S ⊆ V ( G ) is called a general position set of a graph G if no shortest path in G contains more than two vertices from S . Likewise, a subset X ⊆ E ( G ) is an edge general position set if no shortest path contains more than two edges from X . The maximum cardinalities of such sets are known as the general position number and the edge general position number, respectively. This paper focuses on determining these values for the unitary Cayley graph , employing structural methods based on the direct product of graphs. For with p 1 p 2 , we prove that the general position number of is equal to 4 when n = 6, and otherwise equals . Extending to three prime factors, for with p 1 p 2 p 3 , we obtain . Furthermore, the edge general position number of is determined as follows. If p 1 = 2, then it equals ; otherwise, it equals . For the general case , we provide lower bounds.