Independence complexes of well-covered circulantgraphs

Jonathan Earl (Redeemer - NSERC USRA 2014)Kevin Vander Meulen (Redeemer)

Adam Van Tuyl (Lakehead)Catriona Watt (Redeemer - NSERC USRA 2012)

October 2014

E-VM-VT-W Well-covered circulants

Graph Theory I

G = (VG, EG) is a finite simple graph with vertex set VG and edge setEG.

• 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 toinclusion.

Definition (well-covered)

A graph G = (VG, EG) is well-covered if every maximal independent sethas the same cardinality.

E-VM-VT-W Well-covered circulants

Graph Theory II

Example

t3

t4 t2t1t0!!!!!aaaaa

### c

cc t4

t3

t 5 t2t1t0###c

cc

### c

cc

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!

E-VM-VT-W Well-covered circulants

Graph Theory II

Example

t3

t4 t2t1t0!!!!!aaaaa

### c

cc t4

t3

t 5 t2t1t0###c

cc

### c

cc

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!

E-VM-VT-W Well-covered circulants

Graph Theory II

Example

t3

t4 t2t1t0!!!!!aaaaa

### c

cc t4

t3

t 5 t2t1t0###c

cc

### c

cc

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!

E-VM-VT-W 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, . . . , bn2 c}. The circulantgraph Cn(S) is the graph with VG = {0, 1, . . . , n− 1}, such that {a, b} isan 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, Milanic (2014)

E-VM-VT-W Well-covered circulants

Circulants II

• n-cycle Cn is Cn(1).

• n-clique Kn is Cn(1, 2, . . . , bn2 c)

• The circulant C12(1, 3, 4):

E-VM-VT-W 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) havethe 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.

E-VM-VT-W 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} isthe unique maximal facet, or (b), there exists a vertex x such thatlink∆(x) and del∆(x) are vertex decomposable.

(ii) ∆ is shellable if there exists an ordering F1 < F2 < · · · < Ft suchthat for all 1 ≤ j < i ≤ t, there is some x ∈ Fi \ Fj and somek ∈ {1, . . . , j − 1} such that {x} = Fj \ Fk.

(iii) [Reisner’s Criterion] ∆ is Cohen-Macaulay if for all F ∈ ∆,Hi(link∆(F ), k) = 0 for all i < dim link∆(F ). (Here, Hi(−, 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

E-VM-VT-W Well-covered circulants

Independence complexes of well-covered circulant graphs

Problem

Let G be a well-covered circulant. Determine if the pureindependence complex Ind(G) has any richer structure, i.e., vertexdecomposable, shellable, Cohen-Macaulay, or Buchsbaum (ornone).

• Vander Meulen-VT-Watt (Comm. Alg. 2014)

• Earl-Vander Meulen-VT (in progress)

E-VM-VT-W Well-covered circulants

Algebraic connection

Graphs ⇔ SimplicialStanley-Reisner⇔ Commutative

Complexes 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

E-VM-VT-W Well-covered circulants

Results I

Theorem (Brown-Hoshino)

Let n and d be integers with n ≥ 2d ≥ 2. Then Cn(1, 2, . . . , d) iswell-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.

E-VM-VT-W Well-covered circulants

Results I

Theorem (Brown-Hoshino)

Let n and d be integers with n ≥ 2d ≥ 2. Then Cn(1, 2, . . . , d) iswell-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.

E-VM-VT-W Well-covered circulants

Results I

Theorem (Brown-Hoshino)

Let n and d be integers with n ≥ 2d ≥ 2. Then Cn(1, 2, . . . , d) iswell-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.

E-VM-VT-W Well-covered circulants

Results II

Theorem (Brown-Hoshino)

Let n and d be integers with n ≥ 2d + 2 and d ≥ 1. ThenCn(d + 1, d + 2, . . . , bn2 c) is well-covered if and only if n > 3d orn = 2d + 2.

Theorem (Earl-Vander Meulen-VT)

Let n and d be integers with n ≥ 2d + 2 and d ≥ 1. The following areequivalent

(i) Cn(d + 1, d + 2, . . . , bn2 c) is Buchsbaum.

(ii) Cn(d + 1, d + 2, . . . , bn2 c) is well-covered.

(iii) n > 3d or n = 2d + 2.

Furthermore, Cn(d + 1, d + 2, . . . , bn2 c) is vertexdecomposable/shellable/Cohen-Macaulay if and only if n = 2d + 2, ord = 1 and n > 3.

E-VM-VT-W Well-covered circulants

Results III

Theorem (Moussi)

Let G = Cn(S) be the circulant graph with S = {1, . . . , i, . . . , bn2 c} forany 1 ≤ i ≤ bn2 c. Then G is well-covered.

Theorem (Earl-Vander Meulen-VT)

Let G = Cn(S) be the circulant graph with S = {1, . . . , i, . . . , bn2 c} forany 1 ≤ i ≤ bn2 c. Then G is Buchsbaum. Furthermore G is vertexdecomposable/shellable/Cohen-Macaulay if and only gcd(i, n) = 1.

E-VM-VT-W Well-covered circulants

Results IV

Definition

The circulant graph G = Cn(S) is one-paired if there exist an orderedpair 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 soC(12; 3, 2) = C12(3, 9),

E-VM-VT-W Well-covered circulants

Results IV

Theorem (Boros, Gurvich, Milanic)

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 ifand only if n = ab.

(ii) Ind(C(n; a, b)) is Buchsbaum but not Cohen-Macaulay if and onlyif a = 1 and ab < n.

(iii) Ind(C(n; a, b)) is pure but not Buchsbaum if and only if 1 < a andab < n.

E-VM-VT-W 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 reverseimplication holds for Ind(G). Not true for circulant graphs.

Theorem (Earl-Vander Meulen-VT)

(i) The disconnected graph C8(2) is smallest well-covered circulantwhose independence complex is not Buchsbaum. The well-coveredcirculant C10(1, 4) is the smallest connected well-covered graphwith this property.

(ii) The graph C4(1) is the smallest well-covered circulant whoseindependence complex is Buchsbaum but not Cohen-Macaulay.

(iii) The graph C16(1, 4, 8) is the smallest well-covered circulant whoseindependence complex is shellable but not vertex decomposable.

E-VM-VT-W Well-covered circulants

C16(1, 4, 8)

To the best of our knowledge, C16(1, 4, 8) is the first known example ofany 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.

E-VM-VT-W Well-covered circulants

Concluding Remarks

• Moussi’s thesis contains many families of well-covered circulantsthat 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-Macaulaybut not shellable? (I don’t know of any graph G with this property)

E-VM-VT-W Well-covered circulants