Representations for Complete Graphs Minus a Disjoint Union of Paths

A graph \(G\) has a representation modulo \(n\) if there exists an injective map \(f: V(G) \to \{0, 1, \dots, n-1\}\) such that vertices \(u\) and \(v\) are adjacent if and only if \(f(u) – f(v)\) is relatively prime to \(n\). The representation number \(rep(G)\) is the smallest \(n\) such that \(G\) has a representation modulo \(n\). In 2000, Evans, Isaak, and Narayan determined the representation number of a complete graph minus a path. In this paper, we refine their methods and apply them to the family of complete graphs minus a disjoint union of paths.