Depth of powers of binomial edge ideals of complete bipartite graphs

Abstract

Let $G=K_{m,n-m}$ be a complete bipartite graph on $[n]$ with $n-m\ge m$ and J_G its binomial edge ideal in the polynomial ring $S=K[x_1,\dots ,x_n,y_1,\dots ,y_n]$ over a field K. For any $t\ge 2$, it is proved that $\text{depth}(S/J_G^t)$ is equal to n + 1 if m = 1 and 4 if $m\ge 2$.

Keywords:

• Binomial edge ideal
• bipartite graph
• depth

2020 Mathematics Subject Classification:

• 13C15
• 05E40

Acknowledgments

All authors would also like to express their sincere thanks to the referees for their valuable comments.

Additional information

Funding

The second author would like to thank the Natural Science Foundation of Jiangsu Province (No. BK20181427) for a financial support.