A new upper bound for the regularity of gap-free graphs

In this article, we give a new upper bound for the regularity of edge ideals of gap-free graphs, in terms of the their minimal triangulation. Let \(H_U=G\cup F_U\) be a minimal triangulation of a gap-free graph \(G\), for some maximal independent set \(U\) in \(G\). Let \(\mathcal{C}_U\) be the \(3\...

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Bibliographic Details
Published inarXiv.org
Main Authors Nandi, Rimpa, Ramakrishna Nanduri
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 10.09.2021
Subjects
Clutter
Graphs
Regularity
Triangulation
Upper bounds
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Summary:In this article, we give a new upper bound for the regularity of edge ideals of gap-free graphs, in terms of the their minimal triangulation. Let \(H_U=G\cup F_U\) be a minimal triangulation of a gap-free graph \(G\), for some maximal independent set \(U\) in \(G\). Let \(\mathcal{C}_U\) be the \(3\)-uniform clutter of all \(3\)-paths in \(H_U\) which consists of one edge coming from \(F_U\) and another edge coming from \(G\). Then we show that \(\displaystyle \reg(I(G))\leq \reg(I(\C_U))\). As a consequence, we give a general upper bound for the regularity of gap-free graphs. Furthermore, if \(\mathcal{H}\) is the \(3\)-uniform clutter consists of the \(3\)-cliques in \(G\) or in \(F_U\), and the \(3\)-paths in \(G\) which are not \(3\)-cliques in \(H_U\), then \(\reg(I(G))\leq 3\), provided \(\mathcal{H}\) is chordal. This answers partially a question raised by Há, \cite[Problem \(6.3\)]{h14} and by Banerjee, Beyarslan and Há, \cite[Problem \(7.1\)]{bbh19}.
ISSN:2331-8422