#### Vol. 5, No. 4, 2012

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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 $G\left(x\right)$. The vertices of $G\left(x\right)$ 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\le t 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
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