Regularity of symbolic powers of edge ideals of Cameron-Walker graphs

Abstract

A Cameron-Walker graph is a graph for which the matching number and the induced matching number are the same. Assume that G is a Cameron-Walker graph with edge ideal I(G), and let $\text{ind}‐\text{match}(G)$ be the induced matching number of G. It is shown that for every integer $s\ge 1,$ we have the equality $\text{reg}(I{(G)}^{(s)})=2s+\text{ind}‐\text{match}(G)-1,$ where $I{(G)}^{(s)}$ denotes the s-th symbolic power of I(G).

Keywords:

• Cameron-Walker graphs
• Castelnuovo–Mumford regularity
• edge ideal
• symbolic powers

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 13D02
• 05E99

Acknowledgment

The authors thank the referee for careful reading of the paper and for useful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research is partially funded by a Simons Foundation Grant Targeted for Institute of Mathematics, Vietnam Academy of Science and Technology.