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Cover Product and Betti Polynomial of Graphs

Published online by Cambridge University Press:  20 November 2018

Aurora Llamas  and
Josá Martínez–Bernal
Aurora Llamas
Affiliation:
Departamento de Matem´aticas, Cinvestav-IPN, A.P. 14-740, 07000 Máxico D.F.. e-mail: aurora@math.cinvestav.mx, e-mail: jmb@math.cinvestav.mx
Josá Martínez–Bernal
Affiliation:
Departamento de Matem´aticas, Cinvestav-IPN, A.P. 14-740, 07000 Máxico D.F.. e-mail: aurora@math.cinvestav.mx, e-mail: jmb@math.cinvestav.mx
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Abstract

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The cover product of disjoint graphs $G$ and $H$ with fixed vertex covers $C\left( G \right)$ and $C\left( H \right)$, is the graph $G\circledast H$ with vertex set $V\left( G \right)\cup V\left( H \right)$ and edge set

$$E\left( G \right)\,\cup \,E\left( H \right)\,\cup \,\left\{ \left\{ i,\,j \right\}\,:\,i\,\in \,C\left( G \right),\,j\,\in \,C\left( H \right) \right\}.$$

We describe the graded Betti numbers of $G\circledast H$ in terms of those of $G$ and $H$. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph $G$ such that $\text{reg}\,R/I\left( G \right)\,=\,{{\mu }_{s}}\left( G \right)\,+\,k$, where, $I\left( G \right)$ denotes the edge ideal of $G$, $\text{reg}\,\text{R/I}\left( G \right)$ is the Castelnuovo–Mumford regularity of $\text{R/I}\left( G \right)$ and ${{\mu }_{s}}\left( G \right)$ is the induced or strong matching number of $G$; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The $h$-vector of $R/I\left( G\circledast H \right)$ is described in terms of the $h$-vectors of $\text{R/I}\left( G \right)$ and $R/I\left( H \right)$. Furthermore, in a different direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

Keywords

13D0205E45Castelnuovo–Mumford regularitychordal bipartite graphedge idealgraded Betti numberinduced matching numbermonomial ideal
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 320 - 333
Copyright
Copyright © Canadian Mathematical Society 2015

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