Regularity of powers of forests and cycles

By S. Beyarslan, Huy Tài Hà, Tran Nam Trung. 2014

138 citations recorded by the source corpus.

Let G be a graph and let $$I = I(G)$$I=I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of $$I^s$$Is for all $$s \ge 1$$s≥1. In particular, for these classes of graphs, we provide the asymptotic linear function $${{\mathrm{reg}}}(I^s)$$reg(Is) as $$s \gg 0$$s≫0, and the initial value of s starting from which $${{\mathrm{reg}}}(I^s)$$reg(Is) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1be4c60d931dc4298cd1b7bd52b741f1246c2f19

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In Algebra and Number Theory

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Citations in the corpus

Symbolic powers of cover ideal of very well-covered and bipartite graphs (2016, unknown): “Computing and finding bounds for the regularity of powers of a monomial ideal have been studied by a number of researchers (see for example [2], [3], [5], [14]).”
Regularity of Edge Ideals and Their Powers (2017, background): “rms of induced matching. Theorem 4.15. Let Gbe a simple graph and (G) be the induced matching number of G: Then regI(G) = (G) + 2 in the following cases: (a) Gis an n-cycle C n when n2 (mod3) (see [10, 36]); (b) Gis a unicyclic graph with cycle C n when n2 (mod3) and (Gn( G)) = (G) where ( G) is the”
Regularity of powers of bipartite graphs (2016, unknown): “It was shown by Jacques, [17], that if n ≡ 2(mod 3), then reg(I (Cn)) = ν(Cn)+ 2, and Beyarslan, Hà and Trung proved that reg(I (Cn)) = 2s + ν(Cn) − 1 for all s ≥ 2 [5].”