Induced Matchings and the v-Number of Graded Ideals

@article{Grisalde2021InducedMA,
  title={Induced Matchings and the v-Number of Graded Ideals},
  author={Gonzalo Grisalde and Enrique Reyes and Rafael H. Villarreal},
  journal={Mathematics},
  year={2021}
}
We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced… 
View PDF on arXiv

Figures from this paper

References

SHOWING 1-10 OF 60 REFERENCES
Associated primes of monomial ideals and odd holes in graphs
Let G be a finite simple graph with edge ideal I(G). Let I(G)∨ denote the Alexander dual of I(G). We show that a description of all induced cycles of odd length in G is encoded in the associated…
A characterization of well-covered graphs that contain neither 4- nor 5-cycles
TLDR
If G is a connected, well-covered graph containing no 4- nor 5-cycles as subgraphs and G contains an extendable vertex, then G is the disjoint union of edges and triangles together with a restricted set of edges joining extendable vertices.
On well-covered graphs of odd girth 7 or greater
TLDR
It is proved that every connected member G of Gγ=α containing neither C3 nor C5 as a subgraph is a K1, C4, C7 or a corona graph.
Vertex decomposability and regularity of very well-covered graphs
A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph…
A class of planar well-covered graphs with girth four
TLDR
It is shown that all planar 1-well-covered graphs of girth 4 belong to a specific infinite family, and a characterization of this family is given.
Regularity of edge ideals of C4-free graphs via the topology of the lcm-lattice
  • Eran Nevo
  • Computer Science, Mathematics
    J. Comb. Theory, Ser. A
  • 2011
TLDR
This work studies the topology of the lcm-lattice of edge ideals and derives upper bounds on the Castelnuovo-Mumford regularity of the ideals and shows that the second power of the edge ideal has a linear resolution.
Dominating induced matchings of finite graphs and regularity of edge ideals
The regularity of the edge ideal of a finite simple graph G is at least the induced matching number of G and is at most the minimum matching number of G. If G possesses a dominating induced matching,…
A Characterization of Well Covered Graphs of Girth 5 or Greater
TLDR
If G is a connected, well covered graph of girth ≥ 5 and G contains an extendable vertex then G is the disjoint union of edges and 5-cycles together with a restricted set of edges joining these subgraphs.
On the roots of independence polynomials of almost all very well-covered graphs
TLDR
This paper proves that threshold graphs are clique-unique, and demonstrates that the independence polynomial distinguishes well-covered spiders (K"1","n^*,n>=1) among well- covered trees.
On domination and independent domination numbers of a graph
TLDR
Results are obtained for domination number and independent domination number of a graph G, where G does not have an induced subgraph isomorphic to K 1,3, and γ ( G ) = i ( G ).
...
1
2
3
4
5
...