Graphs (Matroids) with k ± ɛ-disjoint spanning trees (bases)

Transcription

1 Graphs (Matroids) with k ± ɛ-disjoint spanning trees (bases) Hong-Jian Lai, Ping Li and Yanting Liang Department of Mathematics West Virginia University Morgantown, WV p. 1/30

2 Notation G: = connected graph p. 2/30

3 Notation G: = connected graph τ(g): = maximum number of edge-disjoint spanning trees of G (spanning tree packing number) p. 2/30

4 Notation G: = connected graph τ(g): = maximum number of edge-disjoint spanning trees of G (spanning tree packing number) Survey Paper: E. M. Palmer, [On the spanning tree packing number of a graph, a survey, Discrete Math. 230 (2001) 13-21]. p. 2/30

5 Packing and Covering Theorems ω(h): = number of connected components of H. p. 3/30

6 Packing and Covering Theorems ω(h): = number of connected components of H. Theorem (Nash-Williams, Tutte [J. London Math. Soc. 36 (1961)]) For a connected graph G, τ(g) k iff X E(G), E X k(ω(g X) 1). p. 3/30

7 Packing and Covering Theorems ω(h): = number of connected components of H. Theorem (Nash-Williams, Tutte [J. London Math. Soc. 36 (1961)]) For a connected graph G, τ(g) k iff X E(G), E X k(ω(g X) 1). a 1 (G):= edge-arboricity, the minimum number of spanning trees whose union equals E(G). p. 3/30

8 Packing and Covering Theorems ω(h): = number of connected components of H. Theorem (Nash-Williams, Tutte [J. London Math. Soc. 36 (1961)]) For a connected graph G, τ(g) k iff X E(G), E X k(ω(g X) 1). a 1 (G):= edge-arboricity, the minimum number of spanning trees whose union equals E(G). Theorem (Nash-Williams, [J. London Math. Soc. 39 (1964)]) a 1 (G) k iff X E(G), X k V (G[X]) ω(g[x]). p. 3/30

9 When does τ(g) = k? By Nash-Williams and Tutte, for a connected G, τ(g) = k if and only if both of the following holds: (i) X E(G), E X k(ω(g X) 1), and (ii) X 0 E(G), E X 0 < (k + 1)(ω(G X) 1). p. 4/30

10 When does τ(g) = k? By Nash-Williams and Tutte, for a connected G, τ(g) = k if and only if both of the following holds: (i) X E(G), E X k(ω(g X) 1), and (ii) X 0 E(G), E X 0 < (k + 1)(ω(G X) 1). What is next? p. 4/30

11 Motivations of Research Suppose that τ(g) < k. What is the minimum number of edges that must be added to G such that the resulting graph has k edge-disjoint spanning trees? p. 5/30

12 Motivations of Research Suppose that τ(g) < k. What is the minimum number of edges that must be added to G such that the resulting graph has k edge-disjoint spanning trees? F (G, k) = this minimum number (Must Added Edges). Determine F (G, k). p. 5/30

13 Motivations of Research Suppose that τ(g) < k. What is the minimum number of edges that must be added to G such that the resulting graph has k edge-disjoint spanning trees? F (G, k) = this minimum number (Must Added Edges). Determine F (G, k). Suppose that τ(g) k. Which edge E(G) has the property that τ(g e) k? p. 5/30

14 Motivations of Research Suppose that τ(g) < k. What is the minimum number of edges that must be added to G such that the resulting graph has k edge-disjoint spanning trees? F (G, k) = this minimum number (Must Added Edges). Determine F (G, k). Suppose that τ(g) k. Which edge E(G) has the property that τ(g e) k? E k (G): = edges with this property (Excessive Edges). Determine E k (G). p. 5/30

15 Example: F (G, K) F (K 3, 2) = 1, F (K 2,t, 2) = 2, and F (P 10, 2) = 3 p. 6/30

16 Example: F (G, K) F (K 3, 2) = 1, F (K 2,t, 2) = 2, and F (P 10, 2) = 3 p. 6/30

17 Example: F (G, K) F (K 3, 2) = 1, F (K 2,t, 2) = 2, and F (P 10, 2) = 3 Every spanning tree must use one of the two edges in the 2-cut. Thus F (G, 2) = 1. p. 6/30

18 Example: E k (G) p. 7/30

19 Example: E k (G) V (G) = 8, E(G) = 18, k = 2 and E(G) 2( V (G) 1) = 4. p. 7/30

20 Example: E k (G) V (G) = 8, E(G) = 18, k = 2 and E(G) 2( V (G) 1) = 4. E 2 (G) = E(K 5 ), (by inspection, or proof postponed). p. 7/30

21 Matroids as a Generalization of Graphs A matroid M consists of a finite set E = E(M) and a collection I(M) of independent subsets of E, satisfying these axioms: (I1) is independent. (I2) Any subset of an independent set is independent. (I3) All maximal independent set in any subset of E have the same cardinality. p. 8/30

22 Matroids as a Generalization of Graphs A matroid M consists of a finite set E = E(M) and a collection I(M) of independent subsets of E, satisfying these axioms: (I1) is independent. (I2) Any subset of an independent set is independent. (I3) All maximal independent set in any subset of E have the same cardinality. Circuits: = minimal dependent sets. p. 8/30

23 Matroids as a Generalization of Graphs A matroid M consists of a finite set E = E(M) and a collection I(M) of independent subsets of E, satisfying these axioms: (I1) is independent. (I2) Any subset of an independent set is independent. (I3) All maximal independent set in any subset of E have the same cardinality. Circuits: = minimal dependent sets. Bases = maximal independent sets p. 8/30

24 Matroids as a Generalization of Graphs A matroid M consists of a finite set E = E(M) and a collection I(M) of independent subsets of E, satisfying these axioms: (I1) is independent. (I2) Any subset of an independent set is independent. (I3) All maximal independent set in any subset of E have the same cardinality. Circuits: = minimal dependent sets. Bases = maximal independent sets Rank of a subset X: r(x) = cardinality of a maximal independent subset in X. p. 8/30

25 Matroids as a Generalization of Graphs M = M(G): Cycle matroid of G on E := E(G). p. 9/30

26 Matroids as a Generalization of Graphs M = M(G): Cycle matroid of G on E := E(G). X E, X := G[X] denotes the induced subgraph. p. 9/30

27 Matroids as a Generalization of Graphs M = M(G): Cycle matroid of G on E := E(G). X E, X := G[X] denotes the induced subgraph. Independent Sets: X E such that X induces an acyclic subgraph p. 9/30

28 Matroids as a Generalization of Graphs M = M(G): Cycle matroid of G on E := E(G). X E, X := G[X] denotes the induced subgraph. Independent Sets: X E such that X induces an acyclic subgraph Circuits (minimal dependent sets): X E such that X is a cycle of G. p. 9/30

29 Matroids as a Generalization of Graphs M = M(G): Cycle matroid of G on E := E(G). X E, X := G[X] denotes the induced subgraph. Independent Sets: X E such that X induces an acyclic subgraph Circuits (minimal dependent sets): X E such that X is a cycle of G. Bases (maximal independent sets): X E such that X is a spanning tree of G. p. 9/30

30 Matroids as a Generalization of Graphs M = M(G): Cycle matroid of G on E := E(G). X E, X := G[X] denotes the induced subgraph. Independent Sets: X E such that X induces an acyclic subgraph Circuits (minimal dependent sets): X E such that X is a cycle of G. Bases (maximal independent sets): X E such that X is a spanning tree of G. Rank r M(G) (X) = V (X) ω(x). p. 9/30

31 Packing and Covering Theorems of Matroids τ(m) : = maximum number disjoint bases of M. p. 10/30

32 Packing and Covering Theorems of Matroids τ(m) : = maximum number disjoint bases of M. If r(m) = 0, then k > 0, τ(m) k. p. 10/30

33 Packing and Covering Theorems of Matroids τ(m) : = maximum number disjoint bases of M. If r(m) = 0, then k > 0, τ(m) k. γ 1 (M): = mimimum number of bases whose union equals E(M). p. 10/30

34 Packing and Covering Theorems of Matroids τ(m) : = maximum number disjoint bases of M. If r(m) = 0, then k > 0, τ(m) k. γ 1 (M): = mimimum number of bases whose union equals E(M). Theorem (Edmonds, [J. Res. Nat. Bur. Standards Sect. B 69B, (1965), 73-77]) Let M be a matroid with r(m) > 0. Each of the following holds. (i) τ(m) k if and only if X E(M), E(M) X k(r(m) r(x)). (ii) γ 1 (M) k if and only if X E(M), X kr(x). p. 10/30

35 Motivations of Research Suppose that τ(m) < k. A (τ k)-extension of M is a matroid M that contains M as a restriction with τ(m ) k. What is the minimum E(M ) E(M) among all (τ k)-extension of M? p. 11/30

36 Motivations of Research Suppose that τ(m) < k. A (τ k)-extension of M is a matroid M that contains M as a restriction with τ(m ) k. What is the minimum E(M ) E(M) among all (τ k)-extension of M? F (M, k) = this minimum number (Must Added Elements). Determine F (M, k). p. 11/30

37 Motivations of Research Suppose that τ(m) < k. A (τ k)-extension of M is a matroid M that contains M as a restriction with τ(m ) k. What is the minimum E(M ) E(M) among all (τ k)-extension of M? F (M, k) = this minimum number (Must Added Elements). Determine F (M, k). Suppose that τ(m) k. Which element e E(M) has the property that τ(m e) k? p. 11/30

38 Motivations of Research Suppose that τ(m) < k. A (τ k)-extension of M is a matroid M that contains M as a restriction with τ(m ) k. What is the minimum E(M ) E(M) among all (τ k)-extension of M? F (M, k) = this minimum number (Must Added Elements). Determine F (M, k). Suppose that τ(m) k. Which element e E(M) has the property that τ(m e) k? E k (M): = elements with this property (Excessive Elements). Determine E k (M). p. 11/30

39 Density of a Subset X E(M) with r(x) > 0, d(x) = X r(x). p. 12/30

40 Density of a Subset X E(M) with r(x) > 0, d(x) = X r(x). d(m) = d(e(m)). p. 12/30

41 Density of a Subset X E(M) with r(x) > 0, d(x) = X r(x). d(m) = d(e(m)). For a (loopless) graph G, X E(G), d(x) = X V (G[X]) ω(g[x]). p. 12/30

42 Density of a Subset X E(M) with r(x) > 0, d(x) = X r(x). d(m) = d(e(m)). For a (loopless) graph G, X E(G), d(x) = X V (G[X]) ω(g[x]). d(g) = d(e(g)). p. 12/30

43 Density of a Subset X E(M) with r(x) > 0, d(x) = X r(x). d(m) = d(e(m)). For a (loopless) graph G, X E(G), d(x) = X V (G[X]) ω(g[x]). d(g) = d(e(g)). Example d(k n ) = n 2, d(k 2,t) = 2t t+1, d(p 10) = 5 3. p. 12/30

44 Density of a Subset X E(M) with r(x) > 0, d(x) = X r(x). d(m) = d(e(m)). For a (loopless) graph G, X E(G), d(x) = X V (G[X]) ω(g[x]). d(g) = d(e(g)). Example d(k n ) = n 2, d(k 2,t) = 2t t+1, d(p 10) = 5 3. d(x) 1, equality holds iff X is independent (G[X] is a forest). p. 12/30

45 Matroid and Graph Contractions For X E(M), M/X is the matroid with rank function r M/X (Y ) = r M (X Y ) r M (X), Y E X. p. 13/30

46 Matroid and Graph Contractions For X E(M), M/X is the matroid with rank function r M/X (Y ) = r M (X Y ) r M (X), Y E X. For X E(G), G/X is obtained from G by identifying the two vertices of each edge in X. The rank function of the cycle matroid of G/X is Y E X, r M(G/X) (Y ) = V (X Y ) ω(x Y ) V (X) + ω(x). p. 13/30

47 Matroid and Graph Contractions For X E(M), M/X is the matroid with rank function r M/X (Y ) = r M (X Y ) r M (X), Y E X. For X E(G), G/X is obtained from G by identifying the two vertices of each edge in X. The rank function of the cycle matroid of G/X is Y E X, r M(G/X) (Y ) = V (X Y ) ω(x Y ) V (X) + ω(x). Example K 8 K 6 K 6 K 4 G K 6 G/K 8 K 6 K 4 G/K 8 K 2 6 K 4 p. 13/30

48 Strength and Fractional Arboricity Reference: [Discrete Applied Math. 40 (1992) ]. p. 14/30

49 Strength and Fractional Arboricity Reference: [Discrete Applied Math. 40 (1992) ]. Strength: η(m) = min{ E X r(m) r(x) min{d(g/x) : r(x) < r(m)}. : r(x) < r(m)} = p. 14/30

50 Strength and Fractional Arboricity Reference: [Discrete Applied Math. 40 (1992) ]. Strength: η(m) = min{ E X r(m) r(x) min{d(g/x) : r(x) < r(m)}. : r(x) < r(m)} = Fractional Arboricity: γ(m) = max{d(x) : r(x) > 0}. p. 14/30

51 Strength and Fractional Arboricity Reference: [Discrete Applied Math. 40 (1992) ]. Strength: η(m) = min{ E X r(m) r(x) min{d(g/x) : r(x) < r(m)}. : r(x) < r(m)} = Fractional Arboricity: γ(m) = max{d(x) : r(x) > 0}. η(m) d(m) γ(m). p. 14/30

52 Strength and Fractional Arboricity Reference: [Discrete Applied Math. 40 (1992) ]. Strength: η(m) = min{ E X r(m) r(x) min{d(g/x) : r(x) < r(m)}. : r(x) < r(m)} = Fractional Arboricity: γ(m) = max{d(x) : r(x) > 0}. η(m) d(m) γ(m). η(g) = η(m(g)), γ(g) = γ(m(g)). p. 14/30

53 Strength and Fractional Arboricity Reference: [Discrete Applied Math. 40 (1992) ]. Strength: η(m) = min{ E X r(m) r(x) min{d(g/x) : r(x) < r(m)}. : r(x) < r(m)} = Fractional Arboricity: γ(m) = max{d(x) : r(x) > 0}. η(m) d(m) γ(m). η(g) = η(m(g)), γ(g) = γ(m(g)). η(g) d(g) γ(g). p. 14/30

54 Strength and Fractional Arboricity Useful Facts ([Discrete Appl. Math. 40 (1992) ]) Each of the following holds. (i) τ(m) = η(g). (ii) γ 1 (M) = γ(g). p. 15/30

55 Strength and Fractional Arboricity Useful Facts ([Discrete Appl. Math. 40 (1992) ]) Each of the following holds. (i) τ(m) = η(g). (ii) γ 1 (M) = γ(g). Theorem (Edmonds, fractional form, [DAM (1992)]) For integers p q > 0, (i) τ(m) p iff M has p bases such that every element q of M is in at most q of them. (ii) γ p iff M has p bases such that every element of q M is in at least q of them. p. 15/30

56 Characterizations Theorem (Catlin, Grossman, Hobbs & HJL, [Discrete Appl. Math. 40 (1992) ]) The following are equivalent. (i) η(m) = d(m). (ii) γ(m) = d(m). (iii) η(m) = γ(m). (iv) η(m) = p, M has p bases such that each element q is in exactly q of them. (v) γ(m) = p, M has p bases such that each element q is in exactly q of them. p. 16/30

57 η-maximal restriction A subset X E(M) is η-maximal if for any Y with X Y E(M), η(m X) > η(m Y ). p. 17/30

58 η-maximal restriction A subset X E(M) is η-maximal if for any Y with X Y E(M), η(m X) > η(m Y ). Example K 8 K 6 K 6 K 4 G K 6 G/K 8 K 6 K 4 G/K 8 K 2 6 K 4 p. 17/30

59 η-maximal restriction A subset X E(M) is η-maximal if for any Y with X Y E(M), η(m X) > η(m Y ). Example K 8 K 6 K 6 K 4 G K 6 G/K 8 K 6 K 4 G/K 8 K 2 6 K 4 Each of K 8, K 6, K 4 is η-maximal. p. 17/30

60 A Decomposition Theorem (i) Theorem Let M be a matroid with r(m) > 0. Then each of the following holds. p. 18/30

61 A Decomposition Theorem (i) Theorem Let M be a matroid with r(m) > 0. Then each of the following holds. (i) There exist an integer m > 0, and an m-tuple (l 1, l 2,..., l m ) of rational numbers such that η(m) = l 1 < l 2 <... < l m = γ(g), and a sequence of subsets J m... J 2 J 1 = E(M); such that for each i with 1 i m, M J i is an η-maximal restriction of M with η(m J i ) = l i. p. 18/30

62 A Decomposition Theorem (ii) (ii) The integer m, the sequences of fractions η(m) = l 1 l 2... l m = γ(m) and subsets J m... J 2 J 1 = E(M) are uniquely determined by M. p. 19/30

63 A Decomposition Theorem (ii) (ii) The integer m, the sequences of fractions η(m) = l 1 l 2... l m = γ(m) and subsets J m... J 2 J 1 = E(M) are uniquely determined by M. Terminologies: p. 19/30

64 A Decomposition Theorem (ii) (ii) The integer m, the sequences of fractions η(m) = l 1 l 2... l m = γ(m) and subsets J m... J 2 J 1 = E(M) are uniquely determined by M. Terminologies: η-spectrum: (l 1, l 2,..., l m ). p. 19/30

65 A Decomposition Theorem (ii) (ii) The integer m, the sequences of fractions η(m) = l 1 l 2... l m = γ(m) and subsets J m... J 2 J 1 = E(M) are uniquely determined by M. Terminologies: η-spectrum: (l 1, l 2,..., l m ). η-decomposition: J m... J 2 J 1 = E(M). p. 19/30

66 Example of The Decomposition p. 20/30

67 Example of The Decomposition m = 2, J 2 = E(K 5 ), J 1 = E(G). p. 20/30

68 Example of The Decomposition m = 2, J 2 = E(K 5 ), J 1 = E(G). i 2 = 5 2, i 1 = 7 4. p. 20/30

69 Example of The Decomposition K 8 K 6 K 6 K 4 G K 6 G/K 8 K 6 K 4 G/K 8 K 2 6 K 4 p. 21/30

70 Example of The Decomposition K 8 K 6 K 6 K 4 G K 6 G/K 8 K 6 K 4 G/K 8 K 2 6 K 4 m = 4, J 4 = E(K 8 ), J 3 = E(K 6 ) E(K 6 ) J 4, J 2 = J 3 E(K 4 ). p. 21/30

71 Example of The Decomposition K 8 K 6 K 6 K 4 G K 6 G/K 8 K 6 K 4 G/K 8 K 2 6 K 4 m = 4, J 4 = E(K 8 ), J 3 = E(K 6 ) E(K 6 ) J 4, J 2 = J 3 E(K 4 ). i 4 = 4, i 3 = 3, i 2 = 2 and i 1 = 5 3. p. 21/30

72 Characterization of Excessive Elements If k > 0 is an integer such that k < ß m = γ(m), then in the η-spectrum, there exists a smallest i j0 such that i j0 > k. J j0 is the η-maximal subset at level k of M. p. 22/30

73 Characterization of Excessive Elements If k > 0 is an integer such that k < ß m = γ(m), then in the η-spectrum, there exists a smallest i j0 such that i j0 > k. J j0 is the η-maximal subset at level k of M. Theorem Let k 2 be an integer. Let M be a graph with τ(m) k. Then each of the following holds. (i) E k (M) = E(M) if and only if η(m) > k. (ii) In general, if η(m) = k and if m > 1, then E k (G) = J 2 equals to the η-maximal subset at level k of M. p. 22/30

74 The Cycle Matroid Case Theorem Let k 2 be an integer, and G be a connected graph with τ(g) k. Let η(m) = l 1 l 2... l m = γ(m) and J m... J 2 J 1 = E(M) denote the η-spectrum and η-decomposition of M(G), respectively. Then each of the following holds. (i) E k (G) = E(G) if and only if η(g) > k. (ii) In general, if η(g) = k and if m > 1, then E k (G) = J 2 equals the η-maximal subset at level k of M(G). p. 23/30

75 The Cycle Matroid Case Theorem Let k 2 be an integer, and G be a connected graph with τ(g) k. Let η(m) = l 1 l 2... l m = γ(m) and J m... J 2 J 1 = E(M) denote the η-spectrum and η-decomposition of M(G), respectively. Then each of the following holds. (i) E k (G) = E(G) if and only if η(g) > k. (ii) In general, if η(g) = k and if m > 1, then E k (G) = J 2 equals the η-maximal subset at level k of M(G). p. 23/30

76 The Cycle Matroid Case Theorem Let k 2 be an integer, and G be a connected graph with τ(g) k. Let η(m) = l 1 l 2... l m = γ(m) and J m... J 2 J 1 = E(M) denote the η-spectrum and η-decomposition of M(G), respectively. Then each of the following holds. (i) E k (G) = E(G) if and only if η(g) > k. (ii) In general, if η(g) = k and if m > 1, then E k (G) = J 2 equals the η-maximal subset at level k of M(G). p. 23/30

77 Example of The Theorem p. 24/30

78 Example of The Theorem m = 2, E 2 (G) = J 2 = E(K 5 ), J 1 = E(G). p. 24/30

79 Example of The Theorem m = 2, E 2 (G) = J 2 = E(K 5 ), J 1 = E(G). i 2 = 5 2, i 1 = 2. p. 24/30

80 What Must Be Added to Have k Disjoint Bases? Let G be a graph, and let F (G, k) denote the minimum number of additional edges that must be added to G to result in a graph G with τ(g ) k. p. 25/30

81 What Must Be Added to Have k Disjoint Bases? Let G be a graph, and let F (G, k) denote the minimum number of additional edges that must be added to G to result in a graph G with τ(g ) k. Since matroids do not have a concept corresponding to vertices, we must formulate the problem differently. p. 25/30

82 What Must Be Added to Have k Disjoint Bases? Let G be a graph, and let F (G, k) denote the minimum number of additional edges that must be added to G to result in a graph G with τ(g ) k. Since matroids do not have a concept corresponding to vertices, we must formulate the problem differently. For a matroid M, a matroid M that contains M as a restriction, and satisfies τ(m ) k is a (τ k)-extension of M. p. 25/30

83 What Must Be Added to Have k Disjoint Bases? Let G be a graph, and let F (G, k) denote the minimum number of additional edges that must be added to G to result in a graph G with τ(g ) k. Since matroids do not have a concept corresponding to vertices, we must formulate the problem differently. For a matroid M, a matroid M that contains M as a restriction, and satisfies τ(m ) k is a (τ k)-extension of M. If τ(m) k, then M = M. p. 25/30

84 What Must Be Added to Have k Disjoint Bases? F (M, k) = min{ E(M ) E(M) : M is a (τ k)-extension of M}. p. 26/30

85 What Must Be Added to Have k Disjoint Bases? F (M, k) = min{ E(M ) E(M) : M is a (τ k)-extension of M}. Theorem Let M be a matroid and let k > 0 be an integer. Each of the following holds. (i) η(m) k if and only if F (M, k) = 0. (ii) If γ(m) k, then F (M, k) = kr(m) E(M). p. 26/30

86 What Must Be Added to Have k Disjoint Bases? F (M, k) = min{ E(M ) E(M) : M is a (τ k)-extension of M}. Theorem Let M be a matroid and let k > 0 be an integer. Each of the following holds. (i) η(m) k if and only if F (M, k) = 0. (ii) If γ(m) k, then F (M, k) = kr(m) E(M). Theorem Let G be a graph. If (edge-arboricity) a 1 (G) k, then F (G, k) = k( V (G) ω(g)) E(G). p. 26/30

87 Application: The Graph Case Theorem (Haas, Theorem 1 of [Ars Combinatoria, 63 (2002), ]) The following are equivalent for a graph G, integers k > 0 and l > 0. (i) E(G) = k( V (G) 1) l and for subgraphs H of G with at least 2 vertices, E(H) k( V (H) 1). (ii) There exists some l edges which when added to G result in a graph that can be decomposed into k spanning trees. p. 27/30

88 Application: The Graph Case Theorem (Haas, Theorem 1 of [Ars Combinatoria, 63 (2002), ]) The following are equivalent for a graph G, integers k > 0 and l > 0. (i) E(G) = k( V (G) 1) l and for subgraphs H of G with at least 2 vertices, E(H) k( V (H) 1). (ii) There exists some l edges which when added to G result in a graph that can be decomposed into k spanning trees. Proof: Either (ii) or E(H) k( V (H) 1) in (i) implies γ(m(g)) k. Hence by our theorem, l = F (G, k) = k( V (G) 1) E(G). p. 27/30

89 What Must Be Added to Have k Disjoint Bases? If k > 0 is an integer such that k < ß m = γ(m), then in the η-spectrum η(m) = l 1 l 2... l m = γ(m), there exists a smallest j(k) such that i(k) := i j(k) k. p. 28/30

90 What Must Be Added to Have k Disjoint Bases? If k > 0 is an integer such that k < ß m = γ(m), then in the η-spectrum η(m) = l 1 l 2... l m = γ(m), there exists a smallest j(k) such that i(k) := i j(k) k. Theorem For integer k > 0, let M be a matroid with τ(m) k and let i(k) denote the smallest i j in η(m) = l 1 l 2... l m = γ(m) such that i(k) k. Then (i) F (M, k) = k(r(m) r(j i(k) ) E(M) J i(k). (ii) F (M, k) = max X E(M) {kr(m/x) M/X }. p. 28/30

91 Application: The Graph Case Theorem (D. Liu, H.-J. Lai and Z.-H. Chen, Theorems 3.4 and 3.10 of [Ars Combinatoria, 93 (2009), ]) Let G be a connected graph with τ(m(g)) k and let i(k) denote the smallest i j in the spectrum of M(G) such that i(k) k. Then (i) F (G, k) = k( V (G) V (G[J i(k) ]) +ω(g[j i(k) ]) 1) E(G) J i(k). (ii) F (G, k) = max Y E(G) {k[ω(g Y ) 1] Y }. p. 29/30

92 Application: The Graph Case Theorem (D. Liu, H.-J. Lai and Z.-H. Chen, Theorems 3.4 and 3.10 of [Ars Combinatoria, 93 (2009), ]) Let G be a connected graph with τ(m(g)) k and let i(k) denote the smallest i j in the spectrum of M(G) such that i(k) k. Then (i) F (G, k) = k( V (G) V (G[J i(k) ]) +ω(g[j i(k) ]) 1) E(G) J i(k). (ii) F (G, k) = max Y E(G) {k[ω(g Y ) 1] Y }. Proof: Apply the theorem to cycle matroids. p. 29/30

93 Thank you! p. 30/30