Maximal Matroids in Weak Order Posets

Let $\cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $\cX$-matroid if each set in $\cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $\cX$-matroid in the weak order poset of all $\cX$-matroids on $E$, and characterizing its rank function when it exists.

1 Introduction 1.1 Unique maximality problem and submodularity conjecture Let X be a family of subsets of a finite set E. We will refer to any matroid on E in which each set in X is a circuit as a X -matroid on E. The set of all X -matroids on E forms a poset under the weak order of matroids in which, for two matroids M 1 and M 2 with the same groundset, we have M 1 M 2 if every independent set in M 1 is independent in M 2 . The main question in this paper is to determine when this poset has a unique maximal element and to characterise this unique maximal matroid when it exists.
Our key tool is the following upper bound on the rank function of any X -matroid on E given in [7]. A proper X -sequence is a sequence S = (X 1 , X 2 , . . . , X k ) of sets in X such that X i ⊂ i−1 j=1 X j for i = 2, . . . , k. For F ⊂ E, let val(F, S) = |F ∪ ( k i=1 X i )| − k. We can use this lemma to derive a sufficient condition for the poset of all X -matroids on E to have a unique maximal element. We need to consider a slightly larger poset. We say that a matroid M on E is X -cyclic if each X ∈ X is a cyclic set in M i.e. for every e ∈ X, there is a circuit C of M with e ∈ C ⊆ X.

Unique maximality problem on graphs
Our primary concern in this paper is the special case when E is the edge set of a graph G and X is the family H G of edge sets of all subgraphs of G which are isomorphic to some member of a given family H of graphs. To simplify terminology we say that a matroid M is a H-matroid on G if it is an H G -matroid on E(G). We will assume throughout that G contains at least one copy of each H ∈ H otherwise we can just consider H\{H}. This implies that the edge sets of any two subgraphs of G which are isomorphic to the same subgraph of a graph H ∈ H will have the same rank in M, but we do not require M to be completely symmetric i.e. the edge sets of every pair of isomorphic subgraphs of G have the same rank. We will simplify notation in the case when H = {H} and refer to a H-matroid on G as a H-matroid on G.
For example, the graphic matroid of K n and the rank two uniform matroid on E(K n ) are both K 3 -matroids on K n .
Chen, Sitharam and Vince have previously considered the unique maximality problem for H-matroids on K n for various graphs H. They announced at a workshop at BIRS in 2015, see [26], that there is a unique maximal K 5 -matroid on K n . Sitharam and Vince subsequently released a preprint [27] which claims to show that there is a unique maximal H-matroid on K n for all graphs H. Unfortunately their claim is false. Pap [21] pointed out that the poset of C 5 -matroids on K n has two maximal elements. We will describe Pap's counterexample, and give other counterexamples to the Sitharam-Vince claim in Section 5.
Our interest in this topic was motivated by the work of Graver, Servatius, and Servatius [11,12] and Whiteley [30] on maximal abstract rigidity matroids, and that of Chan, Sitharam and Vince [26,27] on maximal H-matroids. In two joint papers with Clinch [6,7], we were able to confirm that there is a unique maximal K 5 -matroid on K n and, more importantly, give a good characterisation for the rank function of this matroid. The theory of matroid erections due to Crapo [8] is a key ingredient in our proof technique.
In this paper we will use results on matroid erection from [7] to construct a particular maximal element in the poset of all X -matroids on a set E for certain families X . We will also give examples of pairs (H, G) for which this element is the unique maximal element in the poset of all H-matroids on the graph G, and verify that Conjecture 1.3 holds in each example.

Weakly saturated sequences
The function val X defined in (1) is related to the weak saturation number in extremal graph theory. Let X be a family of subsets of a finite set E, and F 0 ⊆ E. A proper X -sequence (X 1 , X 2 , . . . , X m ) is said to be a weakly X -saturated sequence from F 0 if |X i \ (F 0 ∪ j<i X j )| = 1 for all i with 1 ≤ i ≤ m. We say that E can be constructed by a weakly X -saturated sequence from F 0 if there is a weakly X -saturated sequence S from F 0 with E = F 0 ∪ X∈X X. These sequences were first introduced by Bollobás [4], where he posed the problem of determining the size of a smallest set F 0 from which E can be constructed by a weakly X -saturated sequence. The problem has subsequently been studied by several authors, typically in the case when E is the edge set of the complete k-uniform hypergraphs or the complete bipartite graph, see for example [1,4,15,16,19,22,23]. We will see in Sections 3 and 4 that results on weakly X -saturated sequences can sometimes be used to prove the unique maximality of an X -matroid. However this approach is applicable only when flats of the target matroid are easily described. (The difficulty of deciding uniqueness for when the structure of the flats is more complicated is illustrated by the matroids discussed in Section 6.) The concept of X -matroids was already mentioned implicitly by Kalai [15] and explicitlyby Pikhurko [23], where the goal was to construct an X -matroid to give an upper bound of the length of weakly X -saturated sequences. Our concern in this paper is to gain a more detailed understanding of the set of all X -matroids on a given finite set E.
We close this section by listing notation used throughout the paper. Let M be a matroid on a finite set. Its rank function and the closure operator are denoted by r M and cl M , respectively.
For a graph G, V (G) and E(G) denote its vertex set and its edge set, respectively. Let N G (v) be the set of neighbors of v in G. For F ⊂ E(G), let V (F ) be the set of vertices incident to F and let G[F ] be the graph with vertex set V (F ) and edge set F . Let d F (v) be the number of edges in F incident to a vertex v ∈ V (G), and let N F (v) be the set of For a finite set X, let K(X) be the complete graph with vertex set X. For disjoint finite sets X, Y , let K(X; Y ) be the complete bipartite graph with vertex partition X and Y .

Maximal Matroids and Matroid Elevations
Suppose X be a family of subsets of a finite set E. We first derive a sufficient condition for a given X -matroid on E to be the unique maximal such matroid. We then use results from [7] to construct a maximal element in the poset of all X -matroids on E for certain families X .

A sufficient condition for unique maximality
Recall that a set F in a matroid M is connected if, for every pair of elements e 1 , e 2 ∈ F , there exists a circuit C of M with e 1 , e 2 ∈ C, and that F is a connected component of M if F is either a coloop of M or a maximal connected set in M. It is well known that the set {F 1 , F 2 , . . . , F m } of all connected components partitions the ground set of M and that rank Lemma 2.1. Let X be a family of subsets of a finite set E and M be a loopless X -matroid on E. Suppose that, for every connected flat F of M, there is a proper X -sequence S with r M (F ) = val(F, S). Then val X = r M and M is the unique maximal X -matroid on E.
Proof. Since r M ≤ f for all X -matroids on E by Lemma 1.1, it will suffice to show that, for each F ⊆ E, there is a proper X -sequence S such that r M (F ) = val(F, S).
Suppose, for a contradiction, that this is false for some set F . We may assume that F has been chosen such that r M (F ) is as small as possible and, subject to this condition, |F | is as large as possible. If F is not a flat then r M (F +e) = r M (F ) for some e ∈ E\F and we can now use the maximality of |F | to deduce that there exists a proper X -sequence S such that r M (F + e) = val(F + e, S). By Lemma 1.1 and r M (F ) = r M (F + e), e ∈ X∈S X. Hence, val(F + e, S) = val(F, S) = r M (F + e) = r M (F ). This would contradict the choice of F . Hence F is a flat.
Suppose F is not connected. Then we have Since each F i is a flat, we have X i ⊆ F i for all X i ∈ S i by Lemma 1.1. This implies that the concatenation S = (S 1 , S 2 ) is a proper X -sequence and satisfies This contradicts the choice of F .
Hence F is a connected flat and we can use a hypothesis of the lemma to deduce that there is a proper X -sequence S such that r M (F ) = val(F, S), as required.

Matroid elevations
The truncation of a matroid M 1 = (E, I 1 ) of rank k is the matroid M 0 = (E, I 0 ) of rank k − 1, where I 0 = {I ∈ I 1 : |I| ≤ k − 1}. Crapo [8] defined matroid erection as the 'inverse operation' to truncation. So M 1 is an erection of M 0 if M 0 is the truncation of M 1 . (For technical reasons we also consider M 0 to be a trivial erection of itself.) Note that, although every matroid has a unique truncation, matroids may have several, or no, non-trivial erections.
Crapo [8] showed that the poset of all erections of a matroid M 0 is actually a lattice. It is clear that the trivial erection of M 0 is the unique minimal element in this poset. Since this is a finite lattice, there also exists a unique maximal element called the free erection of M 0 .
A partial elevation of M 0 is any matroid M which can be constructed from M 0 by taking a sequence of erections. A (full) elevation of M 0 is a partial elevation M which has no non-trivial erection. The free elevation of M 0 is the matroid we get from M 0 by recursively constructing a sequence of free erections until we arrive at a matroid which has no non-trivial erection. The set of all partial elevations of M 0 forms a poset P (M 0 ) under the weak order and M 0 is its unique minimal element. Every maximal element of P (M 0 ) will have no non-trivial erection so will be a full elevation of M 0 . Given Crapo's result that the poset of all erections of M 0 is a lattice, it is tempting to conjecture that P (M 0 ) will also be a lattice and that the free elevation of M 0 will be its unique maximal element. But this is false: Brylawski [4] gives a counterexample. The following weaker result is given in [7]. Our next result gives a partial extension of Lemma 2.2 to X -matroids. Given a finite set E and an integer k, let U k (E) be the uniform matroid on E of rank k, i.e. the matroid on E in which a set F ⊆ E is independent if and only if |F | ≤ k. Lemma 2.3. Let E be a finite set, X be a family of subsets of E of size at most s, and M 0 be a maximal matroid in the poset of all X -matroids on E with rank at most s. Suppose that M 0 = U s−1 (E). Then the free elevation of M 0 is a maximal matroid in the poset of all X -matroids on E.
Proof. Let M be the free elevation of M 0 . Since M 0 is a X -matroid and M 0 = U s−1 (E), every set in X is a non-spanning circuit of M 0 . This implies that every partial elevation of M 0 is an X -matroid. In particular, M is an X -matroid.
Lemma 2.2 implies that M is a maximal element in the poset of all partial elevations of M 0 . Let N be an X -matroid on E which is not a partial elevation of M 0 . Let N 0 be the truncation of N to rank s if N has rank at least s, and otherwise let N 0 = N . Then N 0 = M 0 . Since M 0 is a maximal X -matroid on the poset of all X -matroids on E with rank s, N 0 M 0 holds. Hence there exists F ⊆ E with the properties that |F | ≤ s, F is dependent in N 0 and F is independent in M 0 . This implies that F is dependent in N and independent in M so N M. Hence M remains as a maximal element in the poset of all X -matroids on E. Lemma 2.3 can be applied whenever there exists at least one X -matroid M on E since we can truncate M to obtain an X -matroid of rank at most s, and hence the poset of all X -matroids on E with rank at most s will be non-empty. 2 In the next subsection, we given an explicit construction of a maximal X -matroid in the poset of all X -matroids on E with rank at most s whenever X is an s-uniform families.
We close this subsection by stating a useful property of free elevations. We say that an X -matroid M on a finite set E has the X -covering property if every cyclic flat in M is the union of sets in X .

Uniform X -matroids
A family X of sets is k-uniform if each set in X has size k. Given a k-uniform family X , the X -uniform system U X is defined as the pair (E, I X ), where E = X∈X X and We first characterise when U X is a matroid. We say that the k-uniform family X is union-stable if, for any X 1 , X 2 ∈ X and e ∈ X 1 ∩ X 2 , either |( Lemma 2.5. Suppose that X is a k-uniform family. Then U X is a matroid if and only if X is union-stable.
Proof. Let C = X ∪ {C ⊆ E : |C| = k and X ⊆ C for all X ∈ X }. It is straightforward to check U X is a matroid if and only if C satisfies the matroid circuit axioms and that the latter property holds if and only if X is union-stable.
Given an arbitrary k-uniform family X , we construct the union-stable closureX of X by first puttingX = X and then recursively adding (X 1 ∪ X 2 ) − e toX whenever X 1 , X 2 ∈X , |X 1 ∪ X 2 | = k + 1 and e ∈ X 1 ∩ X 2 . It is straightforward to check that the resulting familyX is k-uniform and union-stable and that UX is a maximal matroid in the poset of all X -matroids on E with rank at most k. We can now apply Lemma 2.3 to deduce: Lemma 2.6. Let X be a k-uniform family of sets.Suppose that UX = U k−1 (E). Then the free elevation of UX is a maximal X -matroid on E.
Note that if UX = U k−1 (E) then UX is the unique maximal X -matroid on E but the free-elevation of UX is the free matroid on E i.e. the matroid in which every subset of E is independent.
Suppose that G and H are graphs with |E(H)| = k. We will also assume that every edge of G belongs to a subgraph which is isomorphic to H (we can reduce to this case by deleting all edges of G which do not belong to copies of H). Recall that {H} G denotes the k-uniform family containing all edge sets of copies of H in G. The graph H is said to be union-stable on G if {H} G is union-stable, i.e., for any distinct subgraphs H 1 and H 2 of G isomorphic to H and for any e ∈ E( Examples of union-stable graphs on K n are stars, cycles, complete graphs, and complete bipartite graphs. Lemmas 2.3 and 2.5 now give: is a matroid, then its free elevation is a maximal H-matroid on G.

Weakly Saturated Sequences
Let X be a family of subsets of be a finite set E, and F 0 ⊆ E. Recall that a proper X -sequence (X 1 , X 2 , . . . , X m ) is a weakly X -saturated sequence from F 0 if |X i \ (F 0 ∪ j<i X j )| = 1 for all i with 1 ≤ i ≤ m. We say that a set F ⊆ E can be constructed by a weakly X -saturated sequence from F 0 if there is a weakly X -saturated sequence S from F 0 with F = F 0 ∪ X∈S X. Note that if this is the case then we will have val(F, S) = |F 0 |. We can combine this simple observation with Lemma 2.1 to give several examples of unique maximality.
Lemma 3.1. Let X be a k-uniform family of subsets of a finite set E. Suppose that E can be constructed by a weakly X -saturated sequence from some X 0 ∈ X . Then U k−1 (E) is the unique maximal X -matroid on E and its rank function is val X .
Proof. We denote U = U k−1 (E). Since U is uniform, E is the only connected flat in U and hence, by Lemma 2.1, it will suffice to show that there is a proper X -sequence S such that r U (E) = val X (S, E). By hypothesis, there is a weakly saturated X -sequence S 0 from X 0 to E. Let S be the proper X -sequence obtained by inserting X 0 at the beginning of as required.
The same proof technique can handle a slightly more complicated situation.
Lemma 3.2. Let X be a k-uniform, union-stable family of sets. Suppose that E can be constructed by a weakly X -saturated sequence from some Y ⊆ E with |Y | = k and Y / ∈ X . Then U X is the unique maximal X -matroid on E and its rank function is val X .
Proof. By Lemma 2.1, it will suffice to show that there is a proper X -sequence S such that r U X (F ) = val X (S, F ) for every connected flat in U X . Let F be a connected flat in U X . Then the definition of U X implies that the rank of F is either k or k − 1.
Suppose that the rank of F is k. Then we have F = E. By hypothesis, E can be constructed from Y by a weakly X -saturated sequence S.
Hence we may assume that the rank of F is k − 1. Since F is a flat in U X , every subset of F of size k belongs to X . We will use this fact to define a weakly X -saturated sequence for F . Choose a set F 0 of k − 1 elements in F , and let X e = F 0 ∪ {e} for each e ∈ F \ F 0 . Then each X e ∈ X , and {X e : Applications to matroids on graphs.
Given graphs G and H and subgraphs F 0 , F ⊆ G, we say that F can be constructed by a weakly H-saturated sequence from F 0 if E(F ) can be constructed by a weakly {H} Gsaturated sequence from E(F 0 ). Lemma 3.3. Let H k be the vertex-disjoint union of k copies of K 2 . Then K n can be constructed by a weakly saturated H k -sequence from any copy of H k in K n whenever n ≥ 2k + 1.
Proof. Let H = {e 1 , e 2 , . . . , e k } be a copy of H k in K n for some n ≥ 2k + 1. We show that K n has a weakly saturated H k -sequence starting from H by induction on n. Choose a vertex of v of V (K n )\V (H).
Suppose n = 2k + 1. For each edge f from v to H we can choose a k-matching H f containing f and k − 1 edges of H. Then for each edge g of (K n − v) − E(H) we can choose a k-matching H g containing g and k − 1 edges of H ∪ f ∼v H f . Concatenating H with H f for f ∼ v and then H g for the remaining edges g gives a weakly saturated H k -sequence which constructs K 2k+1 from H. Now suppose n > 2k + 1. By induction, K n − v has a weakly saturated H k -sequence S starting from H. For each edge f from v to H we can choose a k-matching H f containing f and k−1 edges of K n −v. Concatenating H with H f gives the required weakly saturated H k -sequence for K n .
Combining Lemmas 3.1 and 3.3, we immediately obtain: Theorem 3.4. Let H k be the vertex-disjoint union of k copies of K 2 . Then U k−1 (K n ) is the unique maximal H k -matroid on K n for all n ≥ 2k + 1, and f H k is its rank function.
Let P k be the path with k edges. It is straightforward to show that K n can be constructed by a weakly saturated P k -sequence starting from a particular copy of P k in K n whenever n ≥ k + 1. Lemma 3.1 now gives: Theorem 3.5. Let P k be the path of length k. Then U k−1 (K n ) is the unique maximal P k -matroid on K n for all n ≥ k + 1, and f P k is its rank function.
Sitharam and Vince [27] showed that U K 1,3 (K n ) is the unique maximal K 1,3 -matroid on K n for all n ≥ 4. Their result can be deduced from Lemma 3.2 since K 1,3 is unionstable and K n can be constructed by a weakly K 1,3 -saturated sequence starting from a copy of K 3 . We may also deduce that f K 1,3 is the rank function of U K 1,3 (K n ).
It is an open problem to determine whether there is a unique maximal T k -matroid on K n for any fixed tree T k with k edges. The following result gives some information on this poset: it implies that the rank of every T k -matroid on K n is bounded by a quadratic polynomial in k.
Lemma 3.6. Suppose H is a graph with s vertices and minimum degree δ, and M is a is obtained from B i by adding at most δ − 1 edges. This implies that B n has size at most (δ − 1)(n − s + 1) + s−1 2 . The lemma now follows since B n is a base of M.
We close this section with one more application of Lemma 3.2. Consider the graph G 5 in Figure 1. It is straightforward to check that G 5 is union-stable and that K n can be constructed by a weakly saturated G 5 -sequence starting from K 2,3 . Hence U G 5 (K n ) is the unique maximal G 5 -matroid on K n for all n ≥ 5, and f G 5 is its rank function.

Matroids Induced by Submodular Functions
In this section we use weakly saturated sequences and a matroid construction due to Edmonds to give more examples of unique maximal matroids.  Then M f := (E, I f ) is a matroid with rank functionf : 2 E → Z given bŷ We refer to the matroid M f given by Edmond's theorem as the matroid induced by f . Given a set F ⊆ E it is straightforward to check that: F is a circuit in M f if and only if 0 = |F | = f (F ) + 1 and The function f a,b : 2 E(Kn) → Z by f a,b (F ) = a|V (F )| − b is submodular and nondecreasing for all a, b ∈ Z with a ≥ 0 and hence induces a matroid M f a,b (K n ) on E(K n ). These matroids are known as count matroids. It is well known that the cycle matroid of K n is the count matroid M f 1,1 (K n ). Another well-known example is when a = 2 and b = 3, which gives the rigidity matroid of generic frameworks in R 2 . Sitharam and Vince [27] showed that M f 1,1 (K n ) and M f 2,3 (K n ) are the unique maximal K 3 -matroid and K 4 -matroid on K n , respectively. Slightly weaker versions of these results were previously obtained by Graver [11].
We will show that the maximality of both these matroids, as well as several other examples of maximality, follow easily from Lemma 2.1 and Theorem 4.1. We need the following observation on the connected flats of count matroids which follows immediately from (3) and (4).
Let K − n be the graph obtained from K n by removing an edge. (d) M f 2,2 (K n ) is the unique maximal K − 5 -matroid on K n and its rank function is f K −
(e) M f 3,5 (K n ) is the unique maximal K − 6 -matroid on K n and its rank function is f K − 6 .
Proof. In each case M f a,b (K n ) is loopless and is an X-matroid on K n for X = K 3 , K 4 , K − 4 , K − 5 , K − 6 , respectively, by (3). Lemmas 2.1 and 4.2 will now imply that M f a,b (K n ) is the unique maximal X-matroid on K n once we have shown that, for every We will do this by finding a weakly saturated X-sequence which constructs K m from a In cases (a) and (b) we can use the well known fact that, for m ≥ d + 1, K m can be constructed by a weakly saturated K d+2 -sequence starting from the spanning subgraph G with Lemma 2.1 can also be used to extend Theorem 4.3 to matroids on non-complete graphs. For example, if G is a chordal graph, then every connected flat of M f 1,1 (G) is a 2connected chordal graph and we can use Lemma 2.1 and an appropriate weakly saturated K 3 -sequence to deduce that M f 1,1 (G) is the unique maximal K 3 -matroid on G and f K 3 is its rank function.
Our next result gives another example of uniqueness for matroids on non-complete graphs.
Proof. By (3) and (4), each copy of K 2,3 is a circuit in M f 1,0 (K m,n ) and each connected flat is a copy of K s,t for some s ≥ 2, t ≥ 3. By the same argument as in the proof of Theorem 4.3, it will suffice to show that, for any K s,t with s ≥ 2, t ≥ 3, there is a weakly saturated K 2,3 -sequence which constructs K s,t from a subgraph G ⊂ K s,t with |E(G)| = s + t. This follows easily by taking In contrast to this result, we will see in Section 5 that there are two distinct maximal K 2,3 -matroids on K n .
The even cycle matroid is the matroid M on E(K n ), in which a set F is independent if and only if each connected component of the induced subgraph K n [F ] contains at most one cycle, and this cycle is odd if it exists. The rank function of M is given by r M (F ) = |V (F )| − β(F ), where β(F ) denotes the number of bipartite connected components in the graph K n [F ]. We can use this fact to define a modified version of count matroids.
For a, b, c ∈ Z, define g a,b,c : 2 E(Kn) → Z by g a,b,c (F ) = a|V (F )| − bβ(F ) − c. Then g a,b,c is submodular and non-decreasing for all a, b ∈ Z with a ≥ b ≥ 0 since the functions F → |V (F )| and F → |V (F )|−β(F ) are both submodular and non-decreasing. Hence g a,b,c induces a matroid M g a,b,c (K n ) on E(K n ) whenever a ≥ b ≥ 0. We will give examples where M g a,b,c (K n ) is the unique maximal X matroid on K n . We need the following observation on the connected flats of M g a,b,c (K n ) which follows immediately from (3) and (4). Lemma 4.5. Suppose a, b, c ∈ Z with a ≥ b ≥ 0, c ≥ 0, and F ⊆ E(K n ) is a connected flat in M g a,b,c (K n ). Then K n [F ] is either a complete graph with |F | ≥ a|V (F )| − c + 1 or a complete bipartite graph with |F | ≥ a|V (F )| − b − c + 1.
The hypothesis of Lemma 4.5 that c ≥ 0 is needed to ensure that the circuits of M g a,b,c (K n ) induce connected subgraphs of K n , which in turn implies that the same property holds for the connected flats of M g a,b,c (K n ). This is not true when c ≤ −1, for example the disjoint union of two copies of C 4 is both a circuit and a connected flat in M g 1,1,−1 (K n ). Theorem 4.6. (a) The even cycle matroid M g 1,1,0 (K n ) is the unique maximal C 4 -matroid on K n and its rank function is f C 4 . (b) M g 2,1,2 is the unique maximal {K − 5 , K 3,4 }-matroid on K n and its rank function is Proof. In each case, M g a,b,c (K n ) is loopless and is an X -matroid on K n for X = {C 4 } Kn and X = {K − 5 , K 3,4 } Kn , respectively, by (3). Lemmas 2.1 and 4.5 will now imply that M f a,b (K n ) is the unique maximal X -matroid on K n once we have shown that: for every K m ⊆ K n with |E(K m )| ≥ am − c + 1, there is a weakly saturated X -sequence which constructs K m from a subgraph G ⊂ K m with |E(G)| = am − c; for every K s,t ⊆ K n with |E(K s,t )| ≥ am − b − c + 1, there is a weakly saturated X -sequence which constructs K s,t from a subgraph G ⊂ K s,t with |E(G) (a) For m ≥ 4, K m can be constructed by a weakly saturated C 4 -sequence starting from a spanning subgraph G with g 1,1,0 (E(K m )) = m edges by taking For s, t ≥ 2, K s,t can be constructed by a weakly saturated C 4 -sequence starting from a spanning subgraph G with g 1,1,0 (E(K s,t )) = s + t − 1 edges by taking (b) For m ≥ 5, K m can be constructed by a weakly saturated K − 5 -sequence starting from a spanning subgraph G with g 2,1,2 (E(K m )) = 2m − 2 edges by taking For s ≥ 3 and t ≥ 4, K s,t can be constructed by a weakly saturated K 3,4 -sequence starting from a spanning subgraph G with g 2,1,2 (E(K s,t )) = 2(s + t) − 3 edges by taking The matroid M g 2,1,2 (K n ) in Theorem 4.6(b) is the Dilworth truncation of the union of the graphic matroid and the even cycle matroid. It appears in the context of the rigidity of symmetric frameworks in R 2 , see for example [28].
The concept of count matroids has been extended to hypergraphs [22] and to grouplabeled graphs [13]. The technique in this section can be adapted to both settings.
We close this section with a remark on the poset of all {K 4 , K 2,3 }-matroids on K n . It is straightforward to check that M g 1,1,−1 (K n ) is a {K 4 , K 2,3 }-matroid on K n . But we cannot show it is the unique maximal such matroid by using the same proof technique as that of Theorem 4.6 since Lemma 4.5 does not hold when c < 0. In fact, we will see in Theorem 5.4 that M g 1,1,−1 (K n ) is not the unique maximal {K 4 , K 2,3 }-matroid on K n .

Examples of Non-uniquness
We will give three examples of posets of X -matroids on K n in which there is not a unique maximal matroid. We will frequently use the following fact, which follows from the procedure for constructing the free erection of a matroid due to Duke [9], see for example [7, Algorithm 1].
Lemma 5.1. If M 0 is a symmetric matroid on K n , then the free elevation of M 0 is a symmetric matroid on K n . Gyula Pap [21] observed that the cycle matroid of K n and the uniform C 5 -matroid on K n are two distinct maximal C 5 -matroids on K n . We can use Lemma 2.6 to show that Pap's example extends to C k for all k ≥ 5.
Theorem 5.2. There are two distinct maximal C k -matroids on K n for all k ≥ 5 and n ≥ k−1 2 + 2.
Proof. It is straightforward to check that C k is union-stable, and hence U C k (K n ) is a matroid. Consider the free elevation M of U C k (K n ). Lemmas 2.6 and 5.1 imply that M is a maximal C k -matroid on K n and is symmetric. We will show that M contains a circuit Z such that K n [Z] has minimum degree one. To see this, consider two distinct copies X and Y of C k such that X ∩ Y forms a path of length k − 2. By the circuit elimination axiom, (X ∪ Y ) − e contains a circuit Z of M for any e ∈ X ∩ Y . Since the copy of is not a circuit in U C k (K n ), it cannot be a circuit in M. Hence (X ∪ Y ) − e contains a circuit Z such that K n [Z] has minimum degree one. We may now apply Lemma 3.6 with H = Z to deduce that the rank of M is at most k−1 2 . The facts that M is a maximal C k -matroid and the cycle matroid on K n is a C kmatroid of rank n − 1, now imply that there are at least two maximal C k -matroids on K n whenever n ≥ k−1 2 + 2.
In Theorem 4.4, we have seen that the poset of K 2,3 -matroids on K m,n has a unique maximal element. We next show that this statement becomes false if we change the ground set to K n . Theorem 5.3. There are two distinct maximal K 2,3 -matroids on K n for all n ≥ 7.
Proof. Since K 2,3 is union-stable, U K 2,3 (K n ) is a matroid by Lemma 2.7, and its free elevation M is a maximal K 2,3 -matroid on K n and is symmetric by Lemmas 2.6 and 5.1. We will show that U K 2,3 (K n ) has no non-trivial erection and hence M = U K 2,3 (K n ).
We first show that M contains a circuit Z such that K n [Z] has minimum degree one. To see this consider the graphs G 1 and G 2 given in Figure 1. Both are isomorphic to K 2,3 , and hence, by the circuit elimination axiom, the edge set of Figure 1. Since every set of six edges which does not induce a copy K 2,3 is independent in M, the edge set of G 3 is a circuit in M. Since the graph G 4 in Figure 1 is isomorphic to G 3 , the circuit elimination axiom now implies the edge set of G 5 = (G 3 ∪ G 4 ) − v 4 v 5 is dependent in M. Again, since every set of six edges is independent in M unless it induces a copy of K 2,3 , the edge set of G 5 is a circuit in M.
This implies that M is a G 5 -matroid on K n and hence, by Lemma 3.6, the rank of M is at most 6. Since U K 2,3 (K n ) has rank 6, this gives M = U K 2,3 (K n ), and hence U K 2,3 (K n ) is a maximal K 2,3 -matroid on K n .
On the other hand, the bicircular matroid M f 1,0 is a K 2,3 -matroid on K n of rank n. When n ≥ 7, U K 2,3 (K n ) M f 1,0 (K n ) and hence there are at least two maximal K 2,3 -matroids on K n .
Our final example of this section shows that the unique maximality property may not hold even if we restrict our attention to the poset of all partial elevations of a given X -matroid on K n (and hence provides another example, in addition to that given by Brylawski [5], which shows that the free elevation may not be the unique maximal matroid in this poset). Note that the proofs of Theorem 5.2 and 5.3 do not give such an example.   Proof. Let X = {K 4 , K 2,3 } Kn . Since X is 6-uniform and union-stable, U X is a matroid. Hence the free elevation M of U X is a symmetric X -matroid on K n by Lemma 5.1. We will show that M has bounded rank.
Consider the graphs G 3 in Figure 1 and G 6 in Figure 2, and let: does not contain any element in X }.
Claim 5.5. C is the family of circuits of the free erection of U X .
Proof. Let M 1 be the free erection of U X . We first show that M 1 is a {G 3 , G 6 }-matroid by using the argument in the proof of Theorem 5.3. Consider the graphs G 1 and G 2 given in Figure 1. The edge sets of both graphs are circuits in M 1 as they are isomorphic to K 2,3 . By the circuit elimination axiom, the edge set of G 3 is dependent in M 1 . Since every set of six edges which does not induce a copy of K 2,3 or K 4 is independent in M 1 , the edge set of G 3 is a circuit in M 1 . To see that the edge set of G 6 is a circuit in M 1 , consider another copy of K 2,3 , G 1 , which is obtained from G 1 by removing v 4 and adding v 4 with v 1 v 4 , v 2 v 4 . Then we have G 6 ∼ = G 1 ∪ G 1 − v 1 v 5 , and by the same reason as in the previous paragraph, the edge set of G 6 is a circuit in M 1 . Thus, M 1 is a {K 4 , K 2,3 , G 3 , G 6 }-matroid.
One can directly check that C satisfies the circuit axiom for matroids, and that C is the family of circuits of a matroid M of rank 7. By the definition of C , M is the unique maximal matroid in the poset of {K 4 , K 2,3 , G 3 , G 6 }-matroids of rank 7. Since M 1 is a {K 4 , K 2,3 , G 3 , G 6 }-matroid of rank 7, we have M 1 M .
On the other hand, M is a {K 4 , K 2,3 }-matroid of rank 7, and hence it is an erection of U X . Since the free erection M 1 of U X is the unique maximal erection, we have M 1 M . We thus obtain M 1 = M . Claim 5.5 implies that X is the family of all circuits of rank at most 6 in M. We next show that M has bounded rank. Proof. Let D be the edge set of the union of two vertex-disjoint 4-cycles in K n . We split the proof into two cases.
Case 1: Suppose D is dependent in M. Since |D| = 8, Claim 5.6 implies that every proper subset of D is independent, and hence D is a circuit. By Lemma 2.4, the closure D of D in M is the union of copies of K 4 and K 2,3 . Since D does not contain a copy of K 4 or K 2,3 , there is an edge e ∈ D \ D such that D + e contains a circuit C with e ∈ C and C = D + e. Since D + e contains none of K 4 , K 2,3 , G 3 , G 6 and |D + e| = 9, we must have |C| = 8. Observe that any eight-element subset of D + e containing e has a vertex of degree one. Hence, we may use Lemma 3.6 to deduce that M has rank at most 7 2 = 21. Case 2: Suppose D is independent in M. By Theorem 4.4, if F ⊆ E(K n ) induces a bipartite subgraph in K n , then F has rank at most |V (F )| in any K 2,3 -matroid. Using this fact, we compute the rank of the edge set of G 7 in Figure 2. Let F be the edge set of a copy of G 7 in K n and B be a base of F which contains the edge set D of the two disjoint 4-cycles in F . Since D + e induces a bipartite graph with |V (D + e)| + 1 edges for all e ∈ F \D, D + e is dependent. Hence B = D and r M (F ) = |D| = 8.
On the other hand the edge set F of the 5-cycle with a chord in G 7 is independent in M (since it is independent in U X ). These two observations imply that F − e is dependent in M for any e ∈ F \F and hence F − e contains a circuit C of M with 1 ≤ |C\F | ≤ 3. Then the subgraph induced by C has a vertex of degree 1, and we may again use Lemma 3.6 to deduce that M has rank at most 7 2 = 21.
We can now complete the proof by observing that the matroid M g 1,1,−1 (K n ) from Section 4 is a partial elevation of U X (K n ) and has rank n + 1. Since M is a maximal partial elevation of U X (K n ) by Lemma 2.2 and has rank at most 21 by Claim 5.6, M is not the unique maximal partial elevation of U X (K n ) for all n ≥ 21. This completes the proof.
The above proof also implies that there are two distinct maximal {K 4 , K 2,3 }-matroids on K n for all n ≥ 21. The only modification is that we use Lemma 2.6 in the final paragraph to deduce that M is a maximal {K 4 , K 2,3 }-matroid on K n . 6 Matroids from Rigidity, Hyperconnectivity, and Matrix Completion 6.1 Abstract rigidity, cofactor matroids and K d+2 -matroids on K n We are given a generic realisation p : V (K n ) → R d and we would like to know when a subgraph G ⊂ K n is d-rigid i.e., every continuous motion of the vertices of (G, p) which preserves the distance between adjacent pairs of vertices must preserve the distance between all pairs of vertices. The (edge sets of the) d-rigid spanning subgraphs of K n are the bases of a matroid R d (K n ) which is referred to as the d-dimensional generic rigidity matroid. It is well known that R d (K n ) is a K d+2 -matroids on K n and R 1 (K n ) is the cycle matroid of K n . Pollaczek-Geiringer [24] and subsequently Laman [18] showed that is an important open problem in discrete geometry. Graver [11] suggested we may get a better understanding of R d (K n ) by studying the poset of all abstract d-rigidity matroids on K n . This can be defined, using a result of Nguyen [20], as the poset of all K d+2 -matroids on K n of rank dn − d+1 2 . Graver conjectured that R d (K n ) is the unique maximal element in this poset and verified his conjecture for the cases when d = 1, 2. The same proofs yield the slightly stronger results given in Theorem 4.3(a) and (b).
Whiteley [30] showed that Graver's conjecture is false when d ≥ 4 by showing that the cofactor matroid C d−2 d−1 (K n ) from the theory of bivariate splines is an abstract d-rigidity matroid for all d ≥ 1, and that R d (K n ) C d−2 d−1 (K n ) for all d ≥ 4 and sufficiently large n. He offered the revised conjecture that C d−2 d−1 (K n ) is the unique maximal element in the poset of all abstract d-rigidity matroids on K n . We recently verified the case d = 3 of this conjecture in joint work with Clinch [6].
Theorem 6.1 ( [6]). The cofactor matroid C 1 2 (K n ) is the unique maximal K 5 -matroid on K n and f K 5 is its rank function.
The following would be a strengthening of Whiteley's conjecture.
Conjecture 6.2. The cofactor matroid C d−2 d−1 (K n ) is the unique maximal K d+2 -matroid on K n for all d ≥ 1 and f K d+2 is its rank function.

Birigidity and rooted K s,t -matroids on K m,n
Let H be a bipartite graph with bipartition (A, B) and K m,n be a copy of the complete bipartite graph with bipartition (U, W ) where |U | = m and |W | = n. We say that a subgraph H of K m,n is a rooted copy of H in K m,n if there is an isomorphism θ from H to H with θ(A) ⊆ U (and hence θ(B) ⊆ W ). Let {H} * Km,n be the set of all rooted-copies of H in K m,n . A matroid M on K m,n is said to be a rooted H-matroid if it is a {H} * Km,nmatroid. Note that the given ordered bipartition (A, B) of H plays a significant role in this definition -we do not require that an isomorphic image θ(H) of H in K m,n is a circuit in M when θ(A) ⊆ W . On the other hand, if H has an automorphism which maps A onto B, then we will get the same matroid for each ordering of the bipartition of H and this matroid will be equal to the (unrooted) H-matroid on K m,n .

Birigidity matroids
As a primary example of matroids on complete bipartite graphs, we shall introduce the birigidity matroids of Kalai, Nevo, and Novik [17].
Let G = (U ∪W, E) be a bipartite graph, p : U → R k , and q : W → R . We assume that the vertices are ordered as v 1 , v 2 , . . . , v n . We define the (k, )-rigidity matrix of (G, p, q), denoted by R k, (G, p, q), to be the matrix of size |E| × ( |U | + k|W |) in which each vertex in U labels a set of consecutive columns from the first |U | columns, each vertex in W labels a set of k consecutive columns from the last k|W | columns, each row is associated with an edge, and the row labelled by the edge e = u i w j is The generic (k, )-rigidity matroid R k, m,n is the row matroid of R k, (K m,n , p, q) for any generic p and q. It can be checked that the rank of R k, m,n is equal to |U | + k|W | − k , from which it follows that K k+1, +1 is a circuit.
As pointed in [17], R k, m,n coincides with the picture lifting matroids extensively studied by Whiteley [29] when min{k, } = 1. We will show that this matroid is the unique maximal rooted K k+1, +1 -matroid in this case. Theorem 6.3. R k,1 m,n is the unique maximal rooted K k+1,2 -matroid on K m,n .
Proof. Whiteley [29] showed that the picture lifting matroid is the matroid induced by the submodular, non-decreasing function h : 2 E(Kn,m) → Z defined by where U (F ) and W (F ) denote the sets of vertices in U and W , respectively, that are incident to F . Since every connected flat in M h (K m,n ) is a complete bipartite graph K m ,n for some m ≥ 1 and n ≥ 2, we may deduce the theorem from Lemma 2.1 by showing that K m ,n can be constructed by a weakly saturated, rooted K k+1,2 -sequence from a subgraph G with m + kn − k edges. Such a sequence is easily obtained by taking See [1] for details on weakly saturated, rooted K s,t -sequences in K m,n . Lemma 2.1 also tells us that the rank function of R k,1 m,n is determined by proper, rooted K k+1,2 -sequences. We conjecture that this extends to R k, m,n for all k, ≥ 1.
Conjecture 6.4. R k, m,n is the unique maximal rooted K k+1, +1 -matroid on K m,n and its rank function is given by r(F ) = min{val(F, S) : S is a proper, rooted K k+1, +1 -sequence in K m,n } (F ⊆ E(K n,m )).
The special case of this conjecture for R 2,2 m,n is equivalent to a conjecture on the rank function of R 2,2 m,m given in [14,Section 8]. Bernstein [2] gave an NP-type combinatorial characterization for independence in R 2,2 m,n , but giving a co-NP-type characterization remains open even for R 2,2 m,n . Although identifying the maximal K 3,3 -matroids on K m,n is a difficult problem, we can confirm Conjecture 1.3 for K 3,3 -matroids.
Theorem 6.5. The following statements are equivalent.
1. There is a unique maximal K 3,3 -matroid on K m,n .
We will sketch a proof of Theorem 6.5 after Theorem 6.9 below (which gives an analogous result for {K 4 , K 3,3 }-matroids on K n ).

Hyperconnectivity matroids, matrix completion and
{K d , K s,t }-matroids on K n Let p : V (K n ) → R d be a generic map. We assume that the vertices are ordered as V (K n ) = {v 1 , v 2 , . . . , v n }. Kalai [15] defined the d-hyperconnectivity matroid, H d n , to be the row matroid of the matrix of size |E| × d|V | in which each vertex labels a set of d consecutive columns, each row is associated with an edge, and the row labelled by the He showed that, when n ≥ 2d + 2, this matroid is a {K d+2 , K d+1,d+1 }-matroid of rank dn − d+1 2 . As a variant of H d n , Kalai [15] also introduced the matroid I d n , which is the row matroid of the |E| × d|V | matrix with rows instead of (5). He showed that, when n ≥ 2d + 2 and d ≥ 2, I d n is a K d+1,d+1 -matroid on K n of rank dn − d 2 . In the special case when d = 2, this rank constraint implies that I d n is a {K 5 , K 3,3 }-matroid.
The matroids H d n and I d n arise naturally in the context of the rank d completion problem for partially filled n × n matrices which are skew-symmetric and symmetric, respectively [4,25]. The restriction of either I d n or H d n to the complete bipartite graph K m,n is the birigidity matroid R d,d m,n . This matroid arises in the context of the rank d completion problem for partially filled m × n matrices [25].
When d = 1, H 1 n is the cycle matroid (and hence is the unique maximal {K 3 , K 2,2 }matroid on K n ) and I 1 n is the even cycle matroid (and hence is the unique maximal K 2,2 -matroid on K n ).
We can find one more example of a {K d , K s,t }-matroid in rigidity theory. Bolker and Roth [3] showed that K d+2,d+2 is a circuit in the d-dimensional rigidity matroid R d (K n ) when d ≥ 3. Hence R d (K n ) is a {K d+2 , K d+2,d+2 }-matroid on K n for all d ≥ 3.
We conjecture that each of H n d , I n d and R n d is the unique maximal matroid in its respective poset. Conjecture 6.6. (a) For n ≥ 2d+2, H d n is the unique maximal {K d+2 , K d+1,d+1 }-matroid on K n and its rank function is f {K d+2 ,K d+1,d+1 } . (b) For d = 2 and n ≥ 6, I 2 n is the unique maximal {K 5 , K 3,3 }-matroid on K n and its rank function is f {K 5 ,K 3,3 } . (c) For d ≥ 3 and n ≥ 2d + 4, R d n is the unique maximal {K d+2 , K d+2,d+2 }-matroid on K n and its rank function is f {K d+2 ,K d+2,d+2 } . Analyzing the poset of all {K 4 , K 3,3 }-matroids on K n is important since these matroids appear in applications such as the rank two completion of partially filled skew-symmetric matrices and partially-filled rectangular matrices [4,25]. We shall prove that, if this poset has a unique maximal element, then the rank function of the maximal element is f {K 4 ,K 3,3 } . This confirms Conjecture 1.3 for {K 4 , K 3,3 }-matroids. We will need two general results for a matroid on the edge set of a graph. The first was proved for the special case of abstract rigidity matroids in [7]. The same proof gives: For the next lemma, we introduce the 0-extension operation. Given a graph G, the 0-extension operation constructs a new graph by adding a new vertex v 0 and two edges v 0 v 1 and v 0 v 2 with distinct v 1 , v 2 ∈ V (G). We say that a matroid M on K n has the 0-extension property if every 0-extension preserves independence in M, i.e. E(G ) is independent if E(G) is independent and G is obtained from G by a 0-extension operation for any subgraphs G, G of K n . Lemma 6.8. Let M be a K 4 -matroid on K n with the 0-extension property. Then, every circuit in M induces a 2-connected subgraph of K n .
Proof. Suppose, for a contradiction, that some circuit C in M does not induce a 2connected subgraph of K n .
We first consider the case when C is connected. Then C can be partitioned into two sets X and Y such that |V (X) ∩ V (Y )| = 1. Let K be the edge set of the complete graph on V (Y ). Since M is a K 4 -matroid, r M (K) ≤ 2|V (Y )| − 3. The fact that X ∪ Y is a circuit now gives r M (X ∪ K) ≤ r M (X) + r M (K) − 1 ≤ |X| + 2|V (Y )| − 4.
We may construct an independent subset of X ∪ K by extending the independent set X using 0-extensions. Let e be an edge in K incident to the vertex in V (X) ∩ V (Y ). Then X + e is independent by the 0-extension property. Repeatedly applying the 0-extension operation, we can extend X + e to an independent set B of size |X| + 1 + 2(|V (Y )| − 2) = |X| + 2|V (Y )| − 3 by adding edges in K. This contradicts the fact that the rank of X ∪ K is at most |X| + 2|V (Y )| − 4.
The case when C is not connected can be proved similarly.
We need one more graph operation. Given a graph G = (V, E), a vertex v 1 ∈ V , the diamond splitting operation at v 1 (with respect to a fixed partition {U 0 , U * , U 1 } of N G (v 1 ) with |U * | = 2) removes the edges between v 1 and the vertices in U 0 , inserts a new vertex v 0 , and inserts new edges v 0 u for all u ∈ U 0 ∪ U * . We say that a matroid M on K n has the diamond splitting property if any diamond splitting operation preserves independence in M. It was shown in [17] that H 2 n has the 0-extension property and the diamond splitting property.
We can now prove our main result on {K 4 , K 3,3 }-matroids.
(a) There is a unique maximal X -matroid on K n .
(b) The free elevation of U X has the 0-extension property and the diamond splitting property.
(c) There is an X -matroid on K n that has the 0-extension property, the diamond splitting property, and the X -covering property.
(d) val X is submodular on E(K n ).
(a) ⇒ (b): Since X is 6-uniform and union-stable, U X (K n ) is a maximal matroid in the poset of all X -matroids on K n of rank at most 6. Clearly U X (K n ) = U 5 (K n ). Lemma 2.3 and (a) now imply that the free elevation of U X (K n ) is the unique maximal X -matroid on K n . Since H 2 n is an X -matroid on K n with the 0-extension property and the diamond splitting property, the free elevation of U X (K n ) also has the 0-extension property and the diamond splitting property. (c) ⇒ (d): Suppose that (c) holds for some X -matroid M on K n . We prove that r M = val X . By Lemma 2.1, it suffices to show that, for each connected flat F ⊆ E(K n ) of M, there is a proper X -sequence S such that r M (F ) = val(F, S). We prove this by induction on the rank of F .
Since M has the 0-extension property, by Lemma 6.8 every circuit in M induces a 2-connected subgraph of K n . Since M is a K 4 -matroid, r M (F ) ≤ 2|V (F )| − 3 and Lemma 6.7 now imply that there exists a base B of F and a vertex v ∈ V (B) such that d B (v) ≤ 2. Let F v and B v be the set of edges in F and B, respectively, which are not incident to v. We first show that To verify (6) we first note that, since M has the 0-extension property, every circuit in M has minimum degree at least three. Since d B (v) ≤ 2, this implies that cl M (B v ) = F v