Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Power of Edge Ideals of Regularity 3 Bipartite
Graphs
Joint work with Arindam Banerjee
October 21, 2014
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
G is a graph without isolated vertices
S denote the polynomial ring on the vertices of G over
some fixed field K .
The edge ideal of G is
I (G ) = (xy : xy is an edge of G ).
For any homogeneous ideal I of S there exists a graded
minimal free resolution
M
M
0→
S(−j)βk,j → · · · →
S(−j)β0,j → I → 0
j
j
Let I is generated by forms of degree p
1
I has linear resolution if
βi,j = 0
2
for all 1 ≤ i ≤ k and all j 6= p + i
.
I has linear presentation if
β1,j = 0
for all j 6= p + 1
.
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Fröberg:
I (G ) linear resolution ⇐⇒ every induced cycle in Ḡ is a
3-cycle.
Nevo-Peeva and Francisco-Hà-Van Tuyl:
I (G ) linear presentation ⇐⇒ Ḡ has no induced 4-cycle.
Natural Question: What happens for I (G )s for s > 1.
Herzog-Hibi-Zheng :
I (G ) linear resolution =⇒ so does I (G )s for each s ≥ 2.
Francisco-Hà-Van Tuyl:
I (G )s linear resolution for some s ≥ 1 =⇒ Ḡ has no
induced 4-cycles.
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Question (Francisco-Hà-Van Tuyl)
If Ḡ has no induced 4-cycle, then is it true that I (G )s has a
linear resolution for all s ≥ 2?
Is not True.
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Question (Francisco-Hà-Van Tuyl)
If Ḡ has no induced 4-cycle, then is it true that I (G )s has a
linear resolution for all s ≥ 2?
Is not True.
Counterexample (Nevo-Peeva)
Let G be a graph such that Ḡ has no induced cycle and the
clique complex of Ḡ is a triangulation of 2-sphere. Then
reg(I (G )) = 4, reg(I (G )2 ) = 5, reg(I (G )3 ) = 6 and
reg(I (G )4 ) = 8.
I (G )2 doesn’t have a linear resolution but I (G )3 and I (G )4
have linear resolutions.
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
In light of these and based on the Mcaulay2 calculations done
by Francisco, the following question was raised by Irena Peeva,
Question (Nevo-Peeva)
Suppose Ḡ has no induced 4 cycle. Is there an integer k s.t.
I (G )s has linear resolution for s ≥ k ?
We know a counterexample for k = 2.
Holds for bipartite graphs for k = 1.
Definition (Bipartite Graph)
A graph is called bipartite if it contains no odd cycle.
Complement of a bipartite graph can not have any
induced cycle of length greater than four.
For bipartite graphs linear presentation implies linear
resolution.
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Restriction: restrict regularity, first case reg(I (G )) = 3.
Question (Nevo-Peeva)
Suppose Ḡ has no induced 4 cycle and reg(I (G )) = 3. Is it
true that I (G )s has linear resolution for s ≥ 2 ?
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Restriction: restrict regularity, first case reg(I (G )) = 3.
Question (Nevo-Peeva)
Suppose Ḡ has no induced 4 cycle and reg(I (G )) = 3. Is it
true that I (G )s has linear resolution for s ≥ 2 ?
Still Open.
Question (-,Banerjee)
Consider a bipartite graph of reg 3. What we can say about
reg(I (G )s ) for s ≥ 2.
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Theorem (Banerjee)
Let {m1 , . . . , mk } be the set of minimal monomial generators
of I (G )s then
reg(I (G )s+1 ) ≤ max {reg(I (G )s+1 : ml ) + 2s, reg(I (G )s )}.
1≤l≤k
Theorem (-, Banerjee)
G is a bipartite graph.
e1 . . . es be a product of (not necessarily distinct)
s-generators of I = I (G ). Then for each i ∈ {1, 2, . . . , s}

(I s+1

Y
s
: e1 . . . es ) =  I 2 : ei :
ej 
j6=i
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Definition
Two vertices u, v (not necessarily distinct) of G are called
2-connected with respect to edge xy if there is a path uxyv in
G.
Theorem (Banerjee)
For an edge xy in G we have
(I (G )2 : xy ) = I (G ) + (uv : u, v are 2-connected w.r.t xy )
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Example
Joint work
with Arindam
Banerjee
The following are 2-connected vertices
u and u (path in G : uxyu)
z and z (path in G : zxyz)
u and t (path in G : uxyt)
z and u (path in G : zxyu)
z and t (path in G : zxyt)
(I (G )2 : xy ) = I (G ) + (u 2 , z 2 , ut, uz, zt)
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Theorem (-, Banerjee)
If
G is a bipartite graph and s ≥ 1.
M be a product of (not necessarily distinct) s-generators
of I (G ).
Then
1
2
(I (G )s+1 : M) is a quadratic squarefree monomial ideal.
The graph G 0 associated to (I (G )s+1 : M) is bipartite on
the same vertex set of G .
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs
Power of
Edge Ideals of
Regularity 3
Bipartite
Graphs
Joint work
with Arindam
Banerjee
Theorem (-, Banerjee)
G is a bipartite graph and s ≥ 1.
M be a product of (not necessarily distinct) s-generators
of I = I (G ).
reg(I (G )) = 3.
Then
1
reg((I (G )s+1 : M)) ≤ 3.
2
reg(I (G )s ) = 2s + 1.
Joint work with Arindam Banerjee
Power of Edge Ideals of Regularity 3 Bipartite Graphs