Regularity of powers of edge ideals
Huy Tài Hà
Tulane University
Joint with Selvi Beyarslan and Trân Nam Trung
Huy Tài Hà
Regularity of powers of edge ideals
1
Problem and Motivation
Asymptotic linearity of regularity
Stabilization index and free constant
2
Literature Reviews
Polynomial ideals
Edge ideals
Known answers
3
Results
Regularity of powers of forests
Regularity of powers of cycles
Methods and general bounds
Huy Tài Hà
Regularity of powers of edge ideals
Asymptotic linearity of regularity
R a standard graded algebra over a field k
m its maximal homogeneous ideal
M a finitely generated graded R-module
end(M) := max{l | Ml 6= 0}
The regularity of M is
reg(M) = max{end(Hmi (M)) + i}.
Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999);
Trung-Wang (2005))
Let R be a standard graded k -algebra, let I ⊆ R be a
homogeneous ideal and let M be a finitely generated graded
R-module. Then reg(I q M) is asymptotically a linear function in
q, i.e., there exist a and b such that for q 0,
reg(I q M) = aq + b.
Huy Tài Hà
Regularity of powers of edge ideals
Asymptotic linearity of regularity
R a standard graded algebra over a field k
m its maximal homogeneous ideal
M a finitely generated graded R-module
end(M) := max{l | Ml 6= 0}
The regularity of M is
reg(M) = max{end(Hmi (M)) + i}.
Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999);
Trung-Wang (2005))
Let R be a standard graded k -algebra, let I ⊆ R be a
homogeneous ideal and let M be a finitely generated graded
R-module. Then reg(I q M) is asymptotically a linear function in
q, i.e., there exist a and b such that for q 0,
reg(I q M) = aq + b.
Huy Tài Hà
Regularity of powers of edge ideals
Asymptotic linearity of regularity
R a standard graded algebra over a field k
m its maximal homogeneous ideal
M a finitely generated graded R-module
end(M) := max{l | Ml 6= 0}
The regularity of M is
reg(M) = max{end(Hmi (M)) + i}.
Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999);
Trung-Wang (2005))
Let R be a standard graded k -algebra, let I ⊆ R be a
homogeneous ideal and let M be a finitely generated graded
R-module. Then reg(I q M) is asymptotically a linear function in
q, i.e., there exist a and b such that for q 0,
reg(I q M) = aq + b.
Huy Tài Hà
Regularity of powers of edge ideals
Stabilization index and free constant
The constant a is well understood (Trung-Wang (2005))
The free constant b and the stabilization index
q0 = min{q 0 | reg(I q M) = aq + b ∀ q ≥ q 0 }
are not known.
Problem
Understand b and q0 from invariants and properties of I
and M
Explicitly compute b and q0 for special classes of ideals
and modules.
Huy Tài Hà
Regularity of powers of edge ideals
Stabilization index and free constant
The constant a is well understood (Trung-Wang (2005))
The free constant b and the stabilization index
q0 = min{q 0 | reg(I q M) = aq + b ∀ q ≥ q 0 }
are not known.
Problem
Understand b and q0 from invariants and properties of I
and M
Explicitly compute b and q0 for special classes of ideals
and modules.
Huy Tài Hà
Regularity of powers of edge ideals
1
Problem and Motivation
Asymptotic linearity of regularity
Stabilization index and free constant
2
Literature Reviews
Polynomial ideals
Edge ideals
Known answers
3
Results
Regularity of powers of forests
Regularity of powers of cycles
Methods and general bounds
Huy Tài Hà
Regularity of powers of edge ideals
Polynomial ideals
R = k [x1 , . . . , xn ] and I ⊆ R a homogeneous ideal.
Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin
(2013): if I is equi-generated then b is related to the
regularity of fibers of certain projection map
Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin
(2014): if I is m-primary then q0 can be related to “partial”
regularity of the Rees algebra of I.
In practice: regularity of fibers of projection maps and “partial”
regularity of Rees algebras are difficult to compute.
Huy Tài Hà
Regularity of powers of edge ideals
Polynomial ideals
R = k [x1 , . . . , xn ] and I ⊆ R a homogeneous ideal.
Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin
(2013): if I is equi-generated then b is related to the
regularity of fibers of certain projection map
Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin
(2014): if I is m-primary then q0 can be related to “partial”
regularity of the Rees algebra of I.
In practice: regularity of fibers of projection maps and “partial”
regularity of Rees algebras are difficult to compute.
Huy Tài Hà
Regularity of powers of edge ideals
Polynomial ideals
R = k [x1 , . . . , xn ] and I ⊆ R a homogeneous ideal.
Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin
(2013): if I is equi-generated then b is related to the
regularity of fibers of certain projection map
Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin
(2014): if I is m-primary then q0 can be related to “partial”
regularity of the Rees algebra of I.
In practice: regularity of fibers of projection maps and “partial”
regularity of Rees algebras are difficult to compute.
Huy Tài Hà
Regularity of powers of edge ideals
Edge ideals of graphs
V = {x1 , . . . , xn } ←→ R = k [x1 , . . . , xn ]
G = (V , E) a simple graph
The edge ideal of G is
I(G) = xi xj {xi , xj } ∈ E .
Example
x1 v
v x5
@
@
x2 @v
x3 v
v x4
@
@
@v x6
I(G) = (x1 x2 , x2 x3 , x2 x4 , x4 x5 , x4 x6 ) ⊆ k [x1 , . . . , x6 ].
Huy Tài Hà
Regularity of powers of edge ideals
Edge ideals of graphs
V = {x1 , . . . , xn } ←→ R = k [x1 , . . . , xn ]
G = (V , E) a simple graph
The edge ideal of G is
I(G) = xi xj {xi , xj } ∈ E .
Example
x1 v
v x5
@
@
x2 @v
x3 v
v x4
@
@
@v x6
I(G) = (x1 x2 , x2 x3 , x2 x4 , x4 x5 , x4 x6 ) ⊆ k [x1 , . . . , x6 ].
Huy Tài Hà
Regularity of powers of edge ideals
Asymptotic linearity of edge ideals
Know: for q ≥ q0 , reg(I(G)q ) = 2q + b.
Problem
Relate q0 and b to combinatorial data of the graph G
For special classes of graphs, compute q0 and b explicitly.
Huy Tài Hà
Regularity of powers of edge ideals
Asymptotic linearity of edge ideals
Know: for q ≥ q0 , reg(I(G)q ) = 2q + b.
Problem
Relate q0 and b to combinatorial data of the graph G
For special classes of graphs, compute q0 and b explicitly.
Huy Tài Hà
Regularity of powers of edge ideals
Known answers
Herzog-Hibi-Zheng (2004) + Fröberg (1990): if the
complement graph Gc is chordal then I(G)q has a linear
resolution for all q ≥ 1; that is b = 0 and q0 = 1
Ferró-Murgia-Olteanu (2012): if I(G) is an initial or final
lexsegment edge ideal then b = 0 and q0 = 1
Alilooee-Banerjee (2014): if G is a bipartite graph and
reg(I(G)) = 3 then b = 1 and q0 = 1.
Problem
Characterize graphs for which b = 0.
Huy Tài Hà
Regularity of powers of edge ideals
Known answers
Herzog-Hibi-Zheng (2004) + Fröberg (1990): if the
complement graph Gc is chordal then I(G)q has a linear
resolution for all q ≥ 1; that is b = 0 and q0 = 1
Ferró-Murgia-Olteanu (2012): if I(G) is an initial or final
lexsegment edge ideal then b = 0 and q0 = 1
Alilooee-Banerjee (2014): if G is a bipartite graph and
reg(I(G)) = 3 then b = 1 and q0 = 1.
Problem
Characterize graphs for which b = 0.
Huy Tài Hà
Regularity of powers of edge ideals
1
Problem and Motivation
Asymptotic linearity of regularity
Stabilization index and free constant
2
Literature Reviews
Polynomial ideals
Edge ideals
Known answers
3
Results
Regularity of powers of forests
Regularity of powers of cycles
Methods and general bounds
Huy Tài Hà
Regularity of powers of edge ideals
Forests and induced matching number
Definition
Let G = (V , E) be a graph.
A matching in G is a collection of pairwise disjoint edges
An induced matching in G is a matching {e1 , . . . , es } ⊆ E
such that these are also the onlySedges in the induced
subgraph of G over the vertices si=1 ei
The induced matching number of G is the maximum size of
an induced matching in G
The graph G is a forest if it contains no cycles.
Huy Tài Hà
Regularity of powers of edge ideals
Induced matching number
Example
Consider the graph G as follows:
x1 v
v x5
@
@
x2 @v
x3 v
v x4
@
@
@v x6
{x1 x2 , x4 x5 } forms a matching, but not an induced
matching in G
The induced matching number of G is 1.
Huy Tài Hà
Regularity of powers of edge ideals
Regularity of powers of forests
Theorem (Beyarslan, —, Trung (2014))
Let G be a forest and let ν denote its induced matching number.
Then b = ν − 1 and q0 = 1. That is, for all q ≥ 1,
reg(I(G)q ) = 2q + ν − 1.
Huy Tài Hà
Regularity of powers of edge ideals
Regularity of powers of cycles
Theorem (Beyarslan, —, Trung (2014))
jnk
Let G be an n-cycle and let ν =
denote its induced
3
matching number. Then b = ν − 1 and q0 = 2. In fact,
ν + 1 if n ≡ 0, 1 (mod 3)
reg(I(G)) =
ν + 2 if n ≡ 2
(mod 3),
and for all q ≥ 2, we have
reg(I(G)q ) = 2q + ν − 1.
Huy Tài Hà
Regularity of powers of edge ideals
General lower bounds
Theorem (Katzman (2006))
Let G be a graph and let ν denote its induced matching
number. Then
reg(I(G)) ≥ ν + 1.
Theorem (Beyarslan, —, Trung (2014))
Let G be a graph and let ν denote its induced matching
number. Then for any q ≥ 1, we have
reg(I(G)q ≥ 2q + ν − 1.
Huy Tài Hà
Regularity of powers of edge ideals
General lower bounds
Theorem (Katzman (2006))
Let G be a graph and let ν denote its induced matching
number. Then
reg(I(G)) ≥ ν + 1.
Theorem (Beyarslan, —, Trung (2014))
Let G be a graph and let ν denote its induced matching
number. Then for any q ≥ 1, we have
reg(I(G)q ≥ 2q + ν − 1.
Huy Tài Hà
Regularity of powers of edge ideals
Methods
Methods: to establish the upper bound
reg(I(G)q ≤ 2q + ν − 1.
G is a forest: let G0 and G00 be induced subgraphs of G
with no edge in common and whose union is G, and
consider ideals of the form J = I(G0 ) + I(G00 )q .
reg(J) ≤ 2q + ν − 1.
G is a cycle: make use of Banerjee’s recent work, which
states that if {M1 , . . . , Ms } are minimal generators of I(G)q
then
reg(I(G)q+1 ) ≤ max{reg(I(G)q ), reg(I(G)q+1 : Mi ) + 2q}.
Huy Tài Hà
Regularity of powers of edge ideals
Methods
Methods: to establish the upper bound
reg(I(G)q ≤ 2q + ν − 1.
G is a forest: let G0 and G00 be induced subgraphs of G
with no edge in common and whose union is G, and
consider ideals of the form J = I(G0 ) + I(G00 )q .
reg(J) ≤ 2q + ν − 1.
G is a cycle: make use of Banerjee’s recent work, which
states that if {M1 , . . . , Ms } are minimal generators of I(G)q
then
reg(I(G)q+1 ) ≤ max{reg(I(G)q ), reg(I(G)q+1 : Mi ) + 2q}.
Huy Tài Hà
Regularity of powers of edge ideals
Methods
Methods: to establish the upper bound
reg(I(G)q ≤ 2q + ν − 1.
G is a forest: let G0 and G00 be induced subgraphs of G
with no edge in common and whose union is G, and
consider ideals of the form J = I(G0 ) + I(G00 )q .
reg(J) ≤ 2q + ν − 1.
G is a cycle: make use of Banerjee’s recent work, which
states that if {M1 , . . . , Ms } are minimal generators of I(G)q
then
reg(I(G)q+1 ) ≤ max{reg(I(G)q ), reg(I(G)q+1 : Mi ) + 2q}.
Huy Tài Hà
Regularity of powers of edge ideals
Hamiltonian paths and cycles
Definition
Let G be a graph.
A Hamiltonian path in G is a path that goes through each
vertex of G exactly once
A Hamiltonian cycle in G is a cycle that contains all the
vertices of G (and thus, each exactly once).
Huy Tài Hà
Regularity of powers of edge ideals
Hamiltonian paths and cycles
Figure: Graph with Hamiltonian path and cycle.
Huy Tài Hà
Regularity of powers of edge ideals
Regularity of graphs with Hamiltonian paths and
cycles
Theorem (Beyarslan, —, Trung (2014))
Let G be a graph over n vertices.
If G contains a Hamiltonian path then
reg(I(G)) ≤
jn + 1k
3
+ 1.
If G contains a Hamiltonian cycle then
jnk
reg(I(G)) ≤
+ 1.
3
Huy Tài Hà
Regularity of powers of edge ideals
Huy Tài Hà
Regularity of powers of edge ideals