Directed Graphs from Exact Covering Systems

Abstract

Given an exact covering system $S = \{a_i$ (mod $d_i$) $: 1 \leq i \leq r\}$, we introduce the corresponding exact covering system digraph (ECSD) $G_S = G(d_1n+a_1, \ldots, d_rn+a_r)$. The vertices of $G_S$ are the integers and the edges are $(n, d_in+a_i)$ for each $n \in \mathbb{Z}$ and for each congruence in the covering system. We study the structure of these directed graphs, which have finitely many components, one cycle per component, as well as indegree 1 and outdegree $r$ at each vertex. We also explore the link between ECSDs that have a single component and non-standard digital representations of integers.