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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
3 1
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