Infinite Matroidal Version of Hall’s Matching Theorem

Hall’s theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv [2] generalized the notion of matchability to a pair of possibly infinite matroids on the same set and gave a condition that is sufficient for the matchability of a given pair (M,W) of finitary matroids, where the matroid M is SCF — a sum of countably many matroids of finite rank. The condition of Aharoni and Ziv is not necessary for matchability. In this paper we give a condition that is necessary for the existence of a matching for any pair of matroids (not necessarily finitary) and is sufficient for any pair (M,W) of finitary matroids, where the matroid M is SCF. 1. Matroids Following Higgs [6] (see also Oxley [10]), we will define matroid as a pair S = (S, ∂̄) where S is a set and ∂̄ is an IE-operator (idempotent-exchange operator) on S. A space is a pair S = (S, ∂̄) where S is a set and ∂̄ : 2 → 2 is an operator on S such that: M1. X ⊆ ∂̄(X) for every X ⊆ S; M2. if X ⊆ Y ⊆ S, then ∂̄(X) ⊆ ∂̄(Y ). If S = (S, ∂̄) is a space, and ∂̄∗ : 2 → 2 is defined by x ∈ ∂̄∗(X) iff x ∈ X or x / ∈ ∂̄(S \ (X ∪ {x})). then S∗ = (S, ∂̄∗) is also a space (the space dual to S). It is easy to see that the space S∗∗ dual to S∗ is equal to S. A space S = (S, ∂̄) is idempotent if M3. ∂̄(∂̄(X)) = ∂̄(X) for every X ⊆ S; 2 JERZY WOJCIECHOWSKI and it is exchange if M4. for every X, Y and p such that X ⊆ Y ⊆ S and p ∈ S \Y , if p ∈ ∂̄(Y ) \ ∂̄(X) then there is x ∈ Y \X with x ∈ ∂̄(Y \ {x} ∪ {p}). It is a straightforward exercise to verify that a space is idempotent if and only if its dual space is exchange. A matroid is a space that is both idempotent and exchange. A matroid S = (S, ∂̄) is finite if S is finite, and it is finitary if M5. for every X ⊆ S and x ∈ S, if x ∈ ∂̄(X) then there is a finite Y ⊆ X such that x ∈ ∂̄(Y ). A finitary matroid is often called an independence space in the literature. Obviously, every finite matroid is finitary. The space dual to a matroid is clearly also a matroid, but the matroid dual to a finitary matroid (called cofinitary) does not have to be finitary. Let S = (S, ∂̄) be a space and let X ⊆ S. If x ∈ ∂̄(X) or Y ⊆ ∂̄(X), then we say that X spans x or X spans Y , respectively. We say that X is spanning in S if X spans S, and that X is independent in S if no x ∈ X is spanned by X \ {x}. Note that X is independent in S if and only if S \X is spanning in the space dual to S. If X is not independent in S, then we say that it is dependent in S. Given a finitary matroid S = (S, ∂̄), let S̄ be the family of subsets of S that are independent in S. Note that (see Oxley [10]): I1. S̄ 6= ∅; I2. if A ∈ S̄ and B ⊆ A, then B ∈ S̄; I3. if I, J ∈ S̄ are finite and |I| = |J |+ 1, then there is an element y ∈ I \ J such that J ∪ {y} ∈ S̄; I4. if A ⊆ S and I ∈ S̄ for every finite I ⊆ A, then A ∈ S̄. Conversely, if S̄ is a family of subsets of a set S satisfying conditions I1–I4, and ∂̄ : 2 → 2 is defined by x ∈ ∂̄(X) iff x ∈ X or there is A ⊆ X such that A ∈ S̄ and A ∪ {x} / ∈ S̄, MATROIDAL VERSION OF HALL’S THEOREM 3 then S = (S, ∂̄) is a finitary matroid and S̄ is equal to the family of subsets of S that are independent in S. Let S = (S, ∂̄) be a space and let X ⊆ S. If X is both spanning and independent in S, then it is said to be a base of S. It is easy to see that X is a base of S if and only if it is maximal in the family of independent sets of S, and if and only if it is minimal in the family of spanning sets. In general, a matroid may have no bases. Example 1.1. Let S0 = Z and ∂̄0(X) = { X if X is finite; S0 otherwise. It is clear that S0 = ( S0, ∂̄0 ) is a matroid with the family of independent sets equal to the family of all finite subsets of S0 and the family of spanning sets equal to the family of infinite subsets of S0. Thus S0 has no bases. However, if S is a finitary matroid, then for every independent X and spanning Y with X ⊆ Y ⊆ S there is a base B of S with X ⊆ B ⊆ Y . It follows immediately that the same is true for cofinitary matroids. If S = (S, ∂̄) is a matroid and for every Y ⊆ X ⊆ S the family of subsets of X that contain Y and are independent in S has a maximal element, then S is called a B-matroid. Any finitary matroid is a B-matroid. Let Z∞ = Z ∪ {−∞,∞} be the set of quasi-integers. If a1, . . . , an ∈ Z∞, then let the sum a1 + · · · + an be the usual sum if a1, . . . , an are all integers, let the sum be ∞ if at least one of them is ∞, and let it be −∞ if none of a1, . . . , an is ∞ but at least one of them is −∞. Note that it follows immediately from the above definition that the operation of addition in Z∞ is commutative and associative. The difference a− b of two quasi-integers a, b means a+(−b); and likewise, for example, a− b+ c−d means a + (−b) + c + (−d), etc. Let Z∞ be ordered in the obvious way. Note that if a, b, c, d ∈ Z∞ satisfy a ≤ c and b ≤ d, then a + b ≤ c + d. Given a set S, let ‖S‖ ∈ Z∞ be the cardinality of S if S is finite, and ‖S‖ = ∞ if S is infinite. Let S = (S, ∂̄) be a matroid. The quasirank of S (denoted r(S)) is the element of Z∞ that is equal to the maximal cardinality of a finite independent set of S if such 4 JERZY WOJCIECHOWSKI a cardinality exists, and it is equal to ∞ otherwise. If r(S) is finite, then S is said to be a finite-rank matroid. It is obvious that a finite-rank matroid is finitary. If S is finitary, then all bases of S have the same cardinality (denoted ρ(S)), and this cardinality is defined to be the rank of S. Let r∗(S) be the quasirank of the matroid dual to S. Assume that S = (S, ∂̄) is a space and X ⊆ S. The restriction of S to X, denoted S|X, is defined to be the space (X, ∂̄′) with ∂̄′ being the restriction of ∂̄ to 2 . The contraction of S to X, denoted S.X, is the dual space to the restriction to X of the space dual to S. Explicitly, S.X = (X, ∂̄′′) with x ∈ ∂̄′′(A) (where A ⊆ X) if and only if x ∈ ∂̄ (A ∪ (S \X)). If S is a matroid, then both S|X and S.X are matroids. If moreover S is either finite, finite-rank, finitary, or is a B-matroid, then both S|X and S.X have the same property. Let S \X = S| (S \X) and S/X = S. (S \X). Let (Si : i ∈ I) be a family of pairwise disjoint sets and (Si : i ∈ I) be a family of spaces with Si = ( Si, ∂̄i ) . The sum of the family (Si : i ∈ I) is defined to be the space S = (S, ∂̄) with S = i∈I Si and x ∈ ∂̄(X) if and only if x ∈ ∂̄i(X ∩ Si) where i ∈ I is such that x ∈ Si. It is easy to see that the following lemma holds. Lemma 1.1. If the space S = (S, ∂̄) is the sum of the family (Si : i ∈ I) of spaces with Si = ( Si, ∂̄i ) for every i ∈ I, and A ⊆ S, then S|A is the sum of the family (Si|Ai : i ∈ I) and S.A is the sum of the family (Si.Ai : i ∈ I), where Ai = A∩Si for every i ∈ I. A matroid S is said to be SCF if it is the sum of a countable family of finite-rank matroids. 2. A matroidal analog of Hall’s Theorem LetM andW be matroids on a set E. Aharoni and Ziv [2] defined the pair (M,W) to be matchable if there is a subset of E that is both spanning in M and independent in W . Such a subset of E will be called a matching in (M,W). The concept of matchability of a pair of matroids on the same set originated as a generalization of a matroidal interpretation of the existence of a matching in a MATROIDAL VERSION OF HALL’S THEOREM 5 bipartite graph. Indeed, if G = (V, E) is a bipartite graph with bipartition V = M ∪W , then let M = (E, ∂̄M ) and W = (E, ∂̄W ) be the matroids on the set of edges E defined by: • y ∈ ∂̄M(X) if and only if there exists x ∈ X such that x and y are incident to the same vertex in M ; • y ∈ ∂̄W(X) if and only if there exists x ∈ X such that x and y are incident to the same vertex in W . It is easy to see that a subset of E is a matching in Γ = (M,W) if and only if it contains a matching in the graph G, implying that Γ is matchable if and only if the graph G is matchable. Let M and W be matroids on a set E. Aharoni and Ziv define a hindrance in (M,W) to be a subset H of E such that H is independent in bothW andM. ∂̄W(H) ) but H is not spanning in M. ∂̄W(H) ) . They prove the following result. Theorem 2.1. Let M and W be matroids on a set E such that M is SCF and W is finitary. If there are no hindrances in (M,W), then (M,W) is matchable. Theorem 2.1 is used by Aharoni and Ziv to prove a special case of the following conjecture, which is the infinite version of Edmond’s theorem, and is attributed to C. Nash-Williams by Aharoni in [2]. Conjecture 2.2. If M and W are finitary matroids on the same set S, then there exists I ⊆ S such that I is independent in both M and W and there is a partition of I as I = H ∪K with ∂̄M (H) ∪ ∂̄W (K) = S. The condition, in Theorem 2.1, that (M,W) does not contain a hindrance is not necessary for matchability. For example, let E = {(i, j) : i ∈ {0, 1} , j ∈ {0, 1, 2, . . . }} , with (i, j) ∈ ∂̄W (A) iff there is i′ ∈ {0, 1} such that (i′, j) ∈ A, 6 JERZY WOJCIECHOWSKI and X ⊆ E being independent in M if and only if it is the set of edges of an acyclic subgraph of the graph G = (V,E) with V = {0, 1, 2, . . . }, (0, j) incident to j and j + 1, and (1, j) incident to j and j + 2, j = 0, 1, 2, . . . . Then H = {(1, j) : j ∈ {0, 1, 2, . . . }} is a hindrance in (M,W) and T = {(0, j) : j ∈ {0, 1, 2, . . . }} is a matching in (M,W). The condition of Aharoni and Ziv resembles the condition in the countable version of Hall’s theorem proved by Podewski and Steffens [11]. Another countable version of Hall’s Theorem with a condition of a somewhat different nature was given by Nash-Williams [8] [9]. A modified version of the theorem of Nash-Williams, with a condition of a similar nature, called μ-admissibility, is proved in [12]. We are going to formulate a matroidal analog of μ-admissibility after some preliminaries. Let M and W be matroids on a set E. Let M and W be disjoint copies of E (say M = E × {0} and W = E × {1}). In an obvious way, M and W can be regarded as matroids on M and W respectively. To simplify notation, we will often identify the elements of M (of W ) with the elements of E when it does not lead to confusion. A string is an injective function with its domain being an ordinal. In particular, the empty set ∅ is a string with domain 0 = ∅. A string f is said to be in a set S if rge f ⊆ S, and it is said to be an α-string if its domain is equal to α. A string in Γ = (M,W) is a string in M ∪W . Given a string f in Γ, let rgeM f = {a ∈ E : (a, 0) ∈ rge f} , rgeW f = {a ∈ E : (a, 1) ∈ rge f} . A string f in Γ is saturated if rgeM fβ ⊆ rgeW fβ for every β ≤ dom f . Let f be a string and β, γ be ordinals with β ≤ γ ≤ dom f . The [β, γ)-segment of f is the string f[β,γ) defined by f[β,γ)(θ) = f(β + θ), MATROIDAL VERSION OF HALL’S THEOREM 7 for all θ with β + θ < γ, that is, f[β,γ) is obtained from f by restricting it to [β, γ) and shifting the domain to start at 0. Given α ≤ dom f , let fα = f[0,α). Assume that f is a string in Γ. The μ-margin μ(f) of f is an element of Z∞ defined by transfinite induction on α = dom f as follows. Let μ(f) = 0 if α = 0, let (1) μ(f) =    μ(fβ) + 1 if f(β) ∈ W and f(β) is not spanned by rgeW fβ in W , μ(fβ)− 1 if f(β) ∈ M and f(β) is not spanned by E \ rgeM f in M, μ(fβ) otherwise when α = β + 1 is a successor ordinal, and μ(f) = liminf β→α μ(fβ) if α is a limit ordinal. We say that Γ is μ-admissible if μ(f) ≥ 0 for every saturated string f in Γ. We will prove the following results. Theorem 2.3. If M and W are arbitrary matroids on the same set and (M,W) is matchable, then it is μ-admissible. Theorem 2.4. Let M and W be matroids on the same set. If M is SCF, W is finitary, and (M,W) is μ-admissible, then it is matchable. 3. Necessity of the condition In this section we are going to prove Theorem 2.3. Let’s start with the following preliminary lemma. Lemma 3.1. Let S = (S, ∂̄S ) be a matroid, a ∈ S, and {S1, S2, S3} be a partition of S \ {a} (allowing the parts to be empty). Let S ′ i = Si ∪ {a}, i = 1, 2, 3, and S1 = (S/S ′ 1) \ S3, S2 = (S/S1) \ S3, S3 = (S/S1) \ S ′ 3.