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