Rooted order on minimal generators of powers of some cover ideals

We define a total order, which we call rooted order, on minimal generating set of $J(P_n)^s$ where $J(P_n)$ is the cover ideal of a path graph on $n$ vertices. We show that each power of a cover ideal of a path has linear quotients with respect to the rooted order. Along the way, we characterize minimal generating set of $J(P_n)^s$ for $s\geq 3$ in terms of minimal generating set of $J(P_n)^2$. We also discuss the extension of the concept of rooted order to chordal graphs. Computational examples suggest that such order gives linear quotients for powers of cover ideals of chordal graphs as well.Comment: 16 pages, 3 figures, accepted for publication in Osaka Journal of Mathematic

ROOTED ORDER ON MINIMAL GENERATORS OF POWERS OF SOME COVER IDEALS

We define a total order, which we call rooted order, on minimal generating set of J(Pn)s where J(Pn) is the cover ideal of a path graph on n vertices. We show that each power of a cover ideal of a path has linear quotients with respect to the rooted order. Along the way, we characterize minimal generating set of J(Pn)s for s ≥ 3 in terms of minimal generating set of J(Pn)2. We also discuss the extension of the concept of rooted order to chordal graphs. Computational examples suggest that such order gives linear quotients for powers of cover ideals of chordal graphs as well

The size of Betti tables of edge ideals arising from bipartite graphs

Let $\operatorname{pd}(I(G))$ and $\operatorname{reg}(I(G))$ respectively denote the projective dimension and the regularity of the edge ideal $I(G)$ of a graph $G$. For any positive integer $n$, we determine all pairs $(\operatorname{pd}(I(G)),\, \operatorname{reg}(I(G)))$ as $G$ ranges over all connected bipartite graphs on $n$ vertices.Comment: 12 page