Representation Numbers and Prague Dimensions for Complete Graphs Minus a Disjoint Union of Paths

Abstract

A graph is representable modulo $n$ if its vertices can be assigned distinct labels from $\{0,1,2,\dots ,n-1\}$ such that the difference of the labels of two vertices is relatively prime to $n$ if and only if the vertices are adjacent. The representation number $\text{rep}(G)$ is the smallest $n$ such that $G$ has a representation modulo $n$. In this paper, we determine the representation number and the Prague dimension (also known as the product dimension) of a complete graph minus a disjoint union of paths.