Independence complexes of well-covered circulant
graphs
Jonathan Earl (Redeemer - NSERC USRA 2014)
Kevin Vander Meulen (Redeemer)
Adam Van Tuyl (Lakehead)
Catriona Watt (Redeemer - NSERC USRA 2012)
October 2014
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Well-covered circulants
Graph Theory I
G = (VG , EG ) is a finite simple graph with vertex set VG and edge set
EG .
• W ⊆ VG is an independent set if e 6⊂ W for all e ∈ EG .
• W is a maximal independent set if W is maximal with respect to
inclusion.
Definition (well-covered)
A graph G = (VG , EG ) is well-covered if every maximal independent set
has the same cardinality.
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Well-covered circulants
Graph Theory II
Example
0t
1t
0t
1t
#
c
#
c
#
c
cct2
4 #
2 t#5#
c
ta
t
!
aa
!
c
!
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aa!
c
t !
ct
#
t
3
4
3
The first graph is well-covered with maximal independent sets
{1, 3}, {2, 4}, {3, 0}, {4, 1}, {0, 2}.
The second graph is not well-covered with maximal independent sets
{0, 2, 4}, {1, 3, 5}, {2, 5}, {3, 0}, {4, 1}.
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Well-covered circulants
Graph Theory II
Example
0t
1t
0t
1t
#
c
#
c
#
c
cct2
4 #
2 t#5#
c
ta
t
!
aa
!
c
!
##
aa!
c
t !
ct
#
t
3
4
3
The first graph is well-covered with maximal independent sets
{1, 3}, {2, 4}, {3, 0}, {4, 1}, {0, 2}.
The second graph is not well-covered with maximal independent sets
{0, 2, 4}, {1, 3, 5}, {2, 5}, {3, 0}, {4, 1}.
Problem
When is G well-covered?
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Well-covered circulants
Graph Theory II
Example
0t
1t
0t
1t
#
c
#
c
#
c
cct2
4 #
2 t#5#
c
ta
t
!
aa
!
c
!
##
aa!
c
t !
ct
#
t
3
4
3
The first graph is well-covered with maximal independent sets
{1, 3}, {2, 4}, {3, 0}, {4, 1}, {0, 2}.
The second graph is not well-covered with maximal independent sets
{0, 2, 4}, {1, 3, 5}, {2, 5}, {3, 0}, {4, 1}.
Problem
When is G well-covered? KNOWN TO BE NP-COMPLETE!
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Well-covered circulants
Circulants I
Recent attacks have been on circulant graphs.
Definition (Circulant graphs)
Let n ≥ 1 be an integer, and let S ⊆ {1, 2, . . . , b n2 c}. The circulant
graph Cn (S) is the graph with VG = {0, 1, . . . , n − 1}, such that {a, b} is
an edge of Cn (S) if and only if |a − b| ∈ S or n − |a − b| ∈ S.
• Hoshino (Ph.D. 2007)
• Brown & Hoshino (2009, 2011)
• Moussi (M.Sc. 2012)
• Boros, Gurvich, Milanič (2014)
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Well-covered circulants
Circulants II
• n-cycle Cn is Cn (1).
c)
• n-clique Kn is Cn (1, 2, . . . , b n
2
• The circulant C12 (1, 3, 4):
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Well-covered circulants
Independence Complexes
Definition (Independence Complex)
The independence complex of the graph G
Ind(G) = {W ⊆ VG | W is an independent set}
• Ind(G) is a simplicial complex
• A simplicial complex is pure if all its facets (maximal faces) have
the same dimension.
Lemma
G is well-covered ⇔ Ind(G) is pure
Consequence: finding well-covered circulants equivalent to finding
independence complexes of circulants that are pure.
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Well-covered circulants
Pure independence complexes
A pure simplicial complex ∆ may have richer structure.
(i) ∆ is vertex decomposable if (a) ∆ is a simplex, i.e., {x1 , . . . , xn } is
the unique maximal facet, or (b), there exists a vertex x such that
link∆ (x) and del∆ (x) are vertex decomposable.
(ii) ∆ is shellable if there exists an ordering F1 < F2 < · · · < Ft such
that for all 1 ≤ j < i ≤ t, there is some x ∈ Fi \ Fj and some
k ∈ {1, . . . , j − 1} such that {x} = Fj \ Fk .
(iii) [Reisner’s Criterion] ∆ is Cohen-Macaulay if for all F ∈ ∆,
H̃i (link∆ (F ), k) = 0 for all i < dim link∆ (F ). (Here, H̃i (−, k)
denotes the i-th reduced simplicial homology group.)
(iv) ∆ is Buchsbaum if link∆ (x) is Cohen-Macaulay for all x ∈ V .
vertex decomposable ⇒ shellable ⇒ Cohen-Macaulay ⇒ Buchsbaum
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Well-covered circulants
Independence complexes of well-covered circulant graphs
Problem
Let G be a well-covered circulant. Determine if the pure
independence complex Ind(G) has any richer structure, i.e., vertex
decomposable, shellable, Cohen-Macaulay, or Buchsbaum (or
none).
• Vander Meulen-VT-Watt (Comm. Alg. 2014)
• Earl-Vander Meulen-VT (in progress)
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Well-covered circulants
Algebraic connection
Graphs
⇔
Stanley-Reisner
⇔
Simplicial
Complexes
Commutative
Algebra
G
Ind(G)
edge ideal I(G)
G well-covered
Ind(G) pure
I(G) unmixed
Buchsbaum
R/I(G) Buchsbaum
Cohen-Macaulay
R/I(G) C-M
shellable
I(G)∨ linear quotients
vertex decomposable
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Well-covered circulants
Results I
Theorem (Brown-Hoshino)
Let n and d be integers with n ≥ 2d ≥ 2. Then Cn (1, 2, . . . , d) is
well-covered if and only if n ≤ 3d + 2 or n = 4d + 3.
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Well-covered circulants
Results I
Theorem (Brown-Hoshino)
Let n and d be integers with n ≥ 2d ≥ 2. Then Cn (1, 2, . . . , d) is
well-covered if and only if n ≤ 3d + 2 or n = 4d + 3.
Theorem (Vander Meulen-VT-Watt)
Let n and d be integers with n ≥ 2d ≥ 2 and let G = Cn (1, 2, . . . , d).
Then the following are equivalent:
(i) Ind(G) is Cohen-Macaulay.
(ii) Ind(G) is shellable.
(iii) Ind(G) is vertex decomposable.
(iv) n ≤ 3d + 2 and n 6= 2d + 2.
If n = 4d + 3 or n = 2d + 2, then Ind(G) is Buchsbaum.
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Well-covered circulants
Results I
Theorem (Brown-Hoshino)
Let n and d be integers with n ≥ 2d ≥ 2. Then Cn (1, 2, . . . , d) is
well-covered if and only if n ≤ 3d + 2 or n = 4d + 3.
Theorem (Vander Meulen-VT-Watt)
Let n and d be integers with n ≥ 2d ≥ 2 and let G = Cn (1, 2, . . . , d).
Then the following are equivalent:
(i) Ind(G) is Cohen-Macaulay.
(ii) Ind(G) is shellable.
(iii) Ind(G) is vertex decomposable.
(iv) n ≤ 3d + 2 and n 6= 2d + 2.
If n = 4d + 3 or n = 2d + 2, then Ind(G) is Buchsbaum.
Proving Ind(G) is not Cohen-Macaulay for n = 4d + 3 is the hard part.
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Well-covered circulants
Results II
Theorem (Brown-Hoshino)
Let n and d be integers with n ≥ 2d + 2 and d ≥ 1. Then
Cn (d + 1, d + 2, . . . , b n2 c) is well-covered if and only if n > 3d or
n = 2d + 2.
Theorem (Earl-Vander Meulen-VT)
Let n and d be integers with n ≥ 2d + 2 and d ≥ 1. The following are
equivalent
(i) Cn (d + 1, d + 2, . . . , b n2 c) is Buchsbaum.
(ii) Cn (d + 1, d + 2, . . . , b n2 c) is well-covered.
(iii) n > 3d or n = 2d + 2.
Furthermore, Cn (d + 1, d + 2, . . . , b n2 c) is vertex
decomposable/shellable/Cohen-Macaulay if and only if n = 2d + 2, or
d = 1 and n > 3.
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Results III
Theorem (Moussi)
Let G = Cn (S) be the circulant graph with S = {1, . . . , î, . . . , b n2 c} for
any 1 ≤ i ≤ b n2 c. Then G is well-covered.
Theorem (Earl-Vander Meulen-VT)
Let G = Cn (S) be the circulant graph with S = {1, . . . , î, . . . , b n2 c} for
any 1 ≤ i ≤ b n2 c. Then G is Buchsbaum. Furthermore G is vertex
decomposable/shellable/Cohen-Macaulay if and only gcd(i, n) = 1.
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Results IV
Definition
The circulant graph G = Cn (S) is one-paired if there exist an ordered
pair of positive integers (a, b) such that ab|n and
S = {d ∈ [n − 1] : a|d and ab - d}.
One-paired circulant denoted G = C(n; a, b).
Example
Let n = 12 and (a, b) = (3, 2). Then S = {3, 9}, and so
C(12; 3, 2) = C12 (3, 9),
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Results IV
Theorem (Boros, Gurvich, Milanič)
The one-paired circulant G = C(n; a, b) is always well-covered.
Theorem (Earl-Vander Meulen-VT)
(i) Ind(C(n; a, b)) is vertex decomposable/shellable/Cohen-Macaulay if
and only if n = ab.
(ii) Ind(C(n; a, b)) is Buchsbaum but not Cohen-Macaulay if and only
if a = 1 and ab < n.
(iii) Ind(C(n; a, b)) is pure but not Buchsbaum if and only if 1 < a and
ab < n.
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Well-covered circulants
Minimal examples
In general, the implications
vertex decomposable ⇒ shellable ⇒ Cohen-Macaulay ⇒ Buchsbaum
are strict.
For many families of graphs, e.g., bipartite, chordal, the reverse
implication holds for Ind(G). Not true for circulant graphs.
Theorem (Earl-Vander Meulen-VT)
(i) The disconnected graph C8 (2) is smallest well-covered circulant
whose independence complex is not Buchsbaum. The well-covered
circulant C10 (1, 4) is the smallest connected well-covered graph
with this property.
(ii) The graph C4 (1) is the smallest well-covered circulant whose
independence complex is Buchsbaum but not Cohen-Macaulay.
(iii) The graph C16 (1, 4, 8) is the smallest well-covered circulant whose
independence complex is shellable but not vertex decomposable.
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Well-covered circulants
C16 (1, 4, 8)
To the best of our knowledge, C16 (1, 4, 8) is the first known example of
any graph G where Ind(G) is shellable but not vertex decomposable.
Computer experiments suggest if
G = C4s (1, s, 2s) with s ≥ 4
then Ind(G) is shellable but not vertex decomposable.
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Well-covered circulants
Concluding Remarks
• Moussi’s thesis contains many families of well-covered circulants
that haven’t been examined.
• Verify our computer experiments about C4s (1, s, 2s)
• Is there a circulant graph G such that Ind(G) is Cohen-Macaulay
but not shellable? (I don’t know of any graph G with this property)
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Well-covered circulants