Directed Graphs from Exact Covering Systems

TL;DR

This paper introduces a new class of directed graphs derived from exact covering systems, analyzing their structure and linking them to non-standard digital representations of integers.

Contribution

It defines the exact covering system digraph (ECSD), studies its structural properties, and explores connections to digital representations of integers.

Findings

ECSDs have finitely many components with one cycle each

Vertices have indegree 1 and outdegree r

Single-component ECSDs relate to non-standard digital representations

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.