Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.4

Directed Graphs from Exact Covering Systems

Dana Neidmann
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801


Given an exact covering system S = {ai (mod di) : 1 ≤ ir}, we introduce the corresponding exact covering system digraph (ECSD) GS = G(d1n+a1, ..., drn+ar). The vertices of GS are the integers and the edges are (n, din + ai) for each nZ 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.

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(Concerned with sequences A002487 A110081.)

Received March 2 2021; revised versions received November 3 2021; January 24 2022; January 31 2022. Published in Journal of Integer Sequences, February 1 2022.

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