Independence complexes of well-covered circulant graphs

Transcription

1 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

2 Graph Theory I G = (V G, E G ) is a finite simple graph with vertex set V G and edge set E G. W V G is an independent set if e W for all e E G. W is a maximal independent set if W is maximal with respect to inclusion. Definition (well-covered) A graph G = (V G, E G ) is well-covered if every maximal independent set has the same cardinality.

3 Graph Theory II Example 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}.

4 Graph Theory II Example 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?

5 Graph Theory II Example 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!

6 Circulants I Recent attacks have been on circulant graphs. Definition (Circulant graphs) Let n 1 be an integer, and let S {1, 2,..., n 2 }. The circulant graph C n (S) is the graph with V G = {0, 1,..., n 1}, such that {a, b} is an edge of C n (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)

7 Circulants II n-cycle C n is C n(1). n-clique K n is C n(1, 2,..., n 2 ) The circulant C 12 (1, 3, 4):

8 Independence Complexes Definition (Independence Complex) The independence complex of the graph G Ind(G) = {W V G 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.

9 Pure independence complexes A pure simplicial complex may have richer structure. (i) is vertex decomposable if (a) is a simplex, i.e., {x 1,..., x n } 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 F 1 < F 2 < < F t such that for all 1 j < i t, there is some x F i \ F j and some k {1,..., j 1} such that {x} = F j \ F k. (iii) [Reisner s Criterion] is Cohen-Macaulay if for all F, H i (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

10 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)

11 Algebraic connection Graphs Simplicial Complexes Stanley-Reisner Commutative Algebra G Ind(G) edge ideal I(G) G well-covered Ind(G) pure I(G) unmixed Buchsbaum Cohen-Macaulay shellable R/I(G) Buchsbaum R/I(G) C-M I(G) linear quotients vertex decomposable

12 Results I Theorem (Brown-Hoshino) Let n and d be integers with n 2d 2. Then C n (1, 2,..., d) is well-covered if and only if n 3d + 2 or n = 4d + 3.

13 Results I Theorem (Brown-Hoshino) Let n and d be integers with n 2d 2. Then C n (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 = C n (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 2d + 2. If n = 4d + 3 or n = 2d + 2, then Ind(G) is Buchsbaum.

14 Results I Theorem (Brown-Hoshino) Let n and d be integers with n 2d 2. Then C n (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 = C n (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 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.

15 Results II Theorem (Brown-Hoshino) Let n and d be integers with n 2d + 2 and d 1. Then C n (d + 1, d + 2,..., n 2 ) 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) C n (d + 1, d + 2,..., n 2 ) is Buchsbaum. (ii) C n (d + 1, d + 2,..., n 2 ) is well-covered. (iii) n > 3d or n = 2d + 2. Furthermore, C n (d + 1, d + 2,..., n 2 ) is vertex decomposable/shellable/cohen-macaulay if and only if n = 2d + 2, or d = 1 and n > 3.

16 Results III Theorem (Moussi) Let G = C n (S) be the circulant graph with S = {1,..., î,..., n 2 } for any 1 i n 2. Then G is well-covered. Theorem (Earl-Vander Meulen-VT) Let G = C n (S) be the circulant graph with S = {1,..., î,..., n 2 } for any 1 i n 2. Then G is Buchsbaum. Furthermore G is vertex decomposable/shellable/cohen-macaulay if and only gcd(i, n) = 1.

17 Results IV Definition The circulant graph G = C n (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) = C 12 (3, 9),

18 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.

19 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 C 8 (2) is smallest well-covered circulant whose independence complex is not Buchsbaum. The well-covered circulant C 10 (1, 4) is the smallest connected well-covered graph with this property. (ii) The graph C 4 (1) is the smallest well-covered circulant whose independence complex is Buchsbaum but not Cohen-Macaulay. (iii) The graph C 16 (1, 4, 8) is the smallest well-covered circulant whose independence complex is shellable but not vertex decomposable.

20 C 16 (1, 4, 8) To the best of our knowledge, C 16 (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 = C 4s (1, s, 2s) with s 4 then Ind(G) is shellable but not vertex decomposable.

21 Concluding Remarks Moussi s thesis contains many families of well-covered circulants that haven t been examined. Verify our computer experiments about C 4s (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)