Irreducible divisor graphs for numerical monoids

Bachman, Dale; Baeth, Nicholas; Edwards, Craig

doi:10.2140/involve.2012.5.449

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Irreducible divisor graphs for numerical monoids

Dale Bachman, Nicholas Baeth and Craig Edwards

Vol. 5 (2012), No. 4, 449–462

DOI: 10.2140/involve.2012.5.449

Abstract

The factorization of an element $x$ from a numerical monoid can be represented visually as an irreducible divisor graph $G (x)$ . The vertices of $G (x)$ are the monoid generators that appear in some representation of $x$ , with two vertices adjacent if they both appear in the same representation. In this paper, we determine precisely when irreducible divisor graphs of elements in monoids of the form $N = 〈 n, n + 1, \dots, n + t 〉$ where $0 \leq t < n$ are complete, connected, or have a maximum number of vertices. Finally, we give examples of irreducible divisor graphs that are isomorphic to each of the $31$ mutually nonisomorphic connected graphs on at most five vertices.

Keywords

numerical monoids, factorization, irreducible divisor graph, graphs

Mathematical Subject Classification 2010

Primary: 13A05, 20M13

Milestones

Received: 7 November 2011

Revised: 7 February 2012

Accepted: 9 February 2012

Published: 14 June 2013

Communicated by Scott Chapman

Authors

Dale Bachman
Department of Mathematics and Computer Science University of Central Missouri W. C. Morris 222 Warrensburg, MO 64093 United States

Nicholas Baeth
Mathematics and Computer Science University of Central Missouri W. C. Morris 222 Warrensburg, MO 64093 United States

Craig Edwards
Department of Mathematics University of Oklahoma Physical Sciences Center 423 Norman, OK 73019 United States

Vol. 5, No. 4, 2012