Abstract
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.
Acknowledgments
The author's research was partially supported by TÜBİTAK, grant no. 118C033. We would like to thank the anonymous referee for his/her helpful comments. After the submission of this paper, taking an entirely different approach, Herzog, Hibi and Moradi [14] independently proved that all powers of the vertex cover ideal of a path graph have linear quotients.
Citation
Nursel Erey. "Rooted order on minimal generators of powers of some cover ideals." Osaka J. Math. 59 (2) 253 - 267, April 2022.
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