Representations for Complete Graphs Minus a Disjoint Union of Paths

Abstract

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.