Cover Product and Betti Polynomial of Graphs

Llamas, Aurora ; Martínez–Bernal, Josá

Canadian mathematical bulletin, Tome 58 (2015) no. 2, pp. 320-333

Résumé

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.

Détail

DOI : 10.4153/CMB-2015-013-3

Mots-clés : 13D02, 05E45, Castelnuovo–Mumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial ideal