Commuting Graphs of Boundedly Generated Semigroups
Abstract
Ara\'ujo, Kinyon and Konieczny (2011) pose several problems concerning the construction of arbitrary commuting graphs of semigroups. We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems. We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs of semigroups in a wider class (containing the class of nilpotent semigroups), is bounded by the minimal number of generators plus two. We also address a problem concerning knit degree.
Cite
@article{arxiv.1710.05250,
title = {Commuting Graphs of Boundedly Generated Semigroups},
author = {Tomer Bauer and Be'eri Greenfeld},
journal= {arXiv preprint arXiv:1710.05250},
year = {2017}
}
Comments
6 pages; Improved Proposition 8 from published version