Vol. 5, No. 4, 2012

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ISSN: 1944-4184 (e-only)
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Irreducible divisor graphs for numerical monoids

Dale Bachman, Nicholas Baeth and Craig Edwards

Vol. 5 (2012), No. 4, 449–462
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,,n + t where 0 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