# A Bound for the Regularity of Powers of Edge Ideals

A Bound for the Regularity of Powers of Edge Ideals Let G be a graph which has no odd cycle of length at most $$2k-1$$ 2 k - 1 ( $$k\ge 2$$ k ≥ 2 ). Assume that I(G) is the edge ideal of G. We prove $$\mathrm{reg}(I(G)^s) \le 2s+$$ reg ( I ( G ) s ) ≤ 2 s + co-chord $$(G)-1$$ ( G ) - 1 , for every integer s with $$1\le s\le k-1$$ 1 ≤ s ≤ k - 1 . Moreover, we show that if G has no odd cycle of lengths 3 and 5, then I(G) has a linear presentation if and only if $$G^c$$ G c is a chordal graph. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Iranian Mathematical Society Springer Journals

# A Bound for the Regularity of Powers of Edge Ideals

, Volume 44 (3) – Jun 4, 2018
5 pages

/lp/springer_journal/a-bound-for-the-regularity-of-powers-of-edge-ideals-FEf9V0ZMQf
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
1017-060X
eISSN
1735-8515
D.O.I.
10.1007/s41980-018-0038-5
Publisher site
See Article on Publisher Site

### Abstract

Let G be a graph which has no odd cycle of length at most $$2k-1$$ 2 k - 1 ( $$k\ge 2$$ k ≥ 2 ). Assume that I(G) is the edge ideal of G. We prove $$\mathrm{reg}(I(G)^s) \le 2s+$$ reg ( I ( G ) s ) ≤ 2 s + co-chord $$(G)-1$$ ( G ) - 1 , for every integer s with $$1\le s\le k-1$$ 1 ≤ s ≤ k - 1 . Moreover, we show that if G has no odd cycle of lengths 3 and 5, then I(G) has a linear presentation if and only if $$G^c$$ G c is a chordal graph.

### Journal

Bulletin of the Iranian Mathematical SocietySpringer Journals

Published: Jun 4, 2018

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