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ON WEAKLY PERFECT ANNIHILATING-IDEAL GRAPHS

Part of: Ordered sets Graph theory

Published online by Cambridge University Press:  11 May 2021

GANESH S. KADU Open the ORCID record for GANESH S. KADU [Opens in a new window] ,
VINAYAK JOSHI Open the ORCID record for VINAYAK JOSHI [Opens in a new window]  and
SAMRUDDHA GONDE Open the ORCID record for SAMRUDDHA GONDE [Opens in a new window]
GANESH S. KADU
Affiliation:
Department of Mathematics, Savitribai Phule Pune University, Pune, India e-mail: ganeshkadu@gmail.com
VINAYAK JOSHI*
Affiliation:
Department of Mathematics, Savitribai Phule Pune University, Pune, India e-mail: vvjoshi@unipune.ac.in
SAMRUDDHA GONDE
Affiliation:
Indian Institute of Science Education and Research, Pune, India e-mail: s.gonde@hotmail.com
*
e-mail: vinayakjoshi111@yahoo.com
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Abstract

We prove that the annihilating-ideal graph of a commutative semigroup with unity is, in general, not weakly perfect. This settles the conjecture of DeMeyer and Schneider [‘The annihilating-ideal graph of commutative semigroups’, J. Algebra469 (2017), 402–420]. Further, we prove that the zero-divisor graphs of semigroups with respect to semiprime ideals are weakly perfect. This enables us to produce a large class of examples of weakly perfect zero-divisor graphs from a fixed semigroup by choosing different semiprime ideals.

Keywords

annihilating-ideal graphsemigroupsemiprime idealweakly perfect graphzero-divisor graph

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 05C15: Coloring of graphs and hypergraphs 06A07: Combinatorics of partially ordered sets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 362 - 372
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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