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Stable value of depth of symbolic powers of edge ideals of graphs

Nguyen Cong Minh, Tran Nam Trung and Thanh Vu

Vol. 329 (2024), No. 1, 147–164
Abstract

Let G be a simple graph on n vertices. We introduce the notion of bipartite connectivity of G, denoted by bc (G) and prove that

lim sdepth (SI(G)(s)) bc (G),

where I(G) denotes the edge ideal of G and S = k [x1, ,xn] is a standard graded polynomial ring over a field k . We further compute the depth of symbolic powers of edge ideals of several classes of graphs, including odd cycles and whisker graphs of complete graphs to illustrate the cases where the above inequality becomes equality.

Dedicated to Professor Ngo Viet Trung on the occasion of his 70th birthday

Keywords
depth of symbolic powers, cycles, stable value of depth
Mathematical Subject Classification
Primary: 13D02
Secondary: 05E40, 13F55
Milestones
Received: 4 September 2023
Revised: 14 April 2024
Accepted: 22 April 2024
Published: 12 June 2024
Authors
Nguyen Cong Minh
Faculty of Mathematics and Informatics
Hanoi University of Science and Technology
Hanoi
Vietnam
Tran Nam Trung
Institute of Mathematics
Vietnam Academy of Science and Technology (VAST)
Hanoi
Vietnam
Thanh Vu
Institute of Mathematics
Vietnam Academy of Science and Technology (VAST)
Hanoi
Vietnam

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