We prove that a biased graph is gainable over the group 𝖹_3 if and only if it contains no minor isomorphic to (4K_2,∅), ± K_3, or -K_4. We develop a theory of "partial groups" that is analogous to that of partial fields, and we use this theory to show that a biased graph is gainable over every non-trivial group if and only if it is gainable over 𝖹_2 and 𝖹_3. From this we derive an independent proof of the theorem due to Gerards that a biased graph is gainable over every non-trivial group if and only if it has no minor isomorphic to (3K_2,∅), ± K_3, or -K_4.

We present a Myhill-Nerode theorem for hypergraphs. The theorem involves an operation which takes two input structures and produces a hypergraph as output. Using this operation, we define a Myhill-Nerode-type equivalence relation and show that if a class of hypergraphs is definable in the counting monadic second-order logic of hypergraphs, then the equivalence relation has finite index. We apply this tool to classes of gain-graphic matroids, and show that if the group Γ is not uniformly locally finite, then the class of Γgain-graphic matroids is not monadically definable. (A group is uniformly locally finite if, for every k, there is a maximum size amongst subgroups generated by at most k elements.) In addition, we define the conviviality graph of a group, and show that if the group Γ has an infinite conviviality graph, then the class of Γgain-graphic matroids is not monadically definable. This will be useful in future constructions.

We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $N$ is an excluded minor of rank at least ten, then $N$ is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that $N$ already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.

We conjecture that the class of frame matroids can be characterised by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterisation for the class of bicircular matroids. The proof does not depend on an excluded-minor characterisation.

Hlineny's Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of F-representable matroids with bounded branch-width (where F is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the class is pigeonhole. An efficiently pigeonhole class has a stronger property, involving an efficiently-computable equivalence relation on subsets of the ground set. We show that Hlineny's Theorem extends to any efficiently pigeonhole class. In a sequel paper, we use these ideas to extend Hlineny's Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, where H is any finite group. We also give a characterisation of the families of hypergraphs that can be described via tree automata: a family is defined by a tree automaton if and only if it has bounded decomposition-width. Furthermore, we show that if a class of matroids has the pigeonhole property, and can be defined in monadic second-order logic, then any subclass with bounded branch-width has a decidable monadic second-order theory.

Let M be a 3-connected matroid and let F be a field. Let A be a matrix over F representing M and let (G,B) be a biased graph representing M. We characterize the relationship between A and (G,B), settling four conjectures of Zaslavsky. We show that for each matrix representation A and each biased graph representation (G,B) of M, A is projectively equivalent to a canonical matrix representation arising from G as a gain graph over F+ or F× realizing B. Further, we show that the projective equivalence classes of matrix representations of M are in one-to-one correspondence with the switching equivalence classes of gain graphs arising from (G,B), except in one degenerate case.

Let G be a simple n ‐vertex graph and c be a coloring of E ( G ) with n colors, where each color class has size at least 2. We prove that ( G , c ) contains a rainbow cycle of length at most ⌈ n 2 ⌉ , which is best possible. Our result settles a special case of a strengthening of the Caccetta‐Häggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.

The class of quasi-graphic matroids recently introduced by Geelen, Gerards, and Whittle generalises each of the classes of frame matroids and lifted-graphic matroids introduced earlier by Zaslavsky. For each biased graph (G,B) Zaslavsky defined a unique lift matroid L(G,B) and a unique frame matroid F(G,B), each on ground set E(G). We show that in general there may be many quasi-graphic matroids on E(G) and describe them all: for each graph G and partition (B,L,F) of its cycles such that B satisfies the theta property and each cycle in L meets each cycle in F, there is a quasi-graphic matroid M(G,B,L,F) on E(G). Moreover, every quasi-graphic matroid arises in this way. We provide cryptomorphic descriptions in terms of subgraphs corresponding to circuits, cocircuits, independent sets, and bases. Equipped with these descriptions, we prove some results about quasi-graphic matroids. In particular, we provide alternate proofs that do not require 3-connectivity of two results of Geelen, Gerards, and Whittle for 3-connected matroids from their introductory paper: namely, that every quasi-graphic matroid linearly representable over a field is either lifted-graphic or frame, and that if a matroid M has a framework with a loop that is not a loop of M then M is either lifted-graphic or frame. We also provide sufficient conditions for a quasi-graphic matroid to have a unique framework. Zaslavsky has asked for those matroids whose independent sets are contained in the collection of independent sets of F(G,B) while containing those of L(G,B), for some biased graph (G,B). Adding a natural (and necessary) non-degeneracy condition defines a class of matroids, which we call biased-graphic. We show that the class of biased-graphic matroids almost coincides with the class of quasi-graphic matroids: every quasi-graphic matroid is biased-graphic, and if M is a biased-graphic matroid that is not quasi-graphic then M is a 2-sum of a frame matroid with one or more lifted-graphic matroids.

Let $M$ be a representable matroid on $n$ elements. We give bounds, in terms of $n$, on the least positive characteristic and smallest field over which $M$ is representable.

Let M be a representable matroid on n elements. We give bounds, in terms of n, on the least positive characteristic and smallest field over which M is representable. Our starting point is given by the following two theorems of Rado [5]. Theorem 1 (Rado, 1957). Let M be a matroid representable over a field K. Then M is representable over a simple algebraic extension of the prime field of K. Theorem 2 (Rado, 1957). Let K be an extension field of Q of degree N , and let M be a matroid representable over K. Then there is a positive integer c such that given any prime p > c there is a positive integer k = k(p) ≤ N such that M is representable over GF(pk). For infinitely many p, k(p) = 1. Together, these two theorems say that if a matroid is linearly representable, then it is representable over a finite field. We ask, given a representable matroid on n elements, how large must such a field be? That is, given an n-element representable matroid M , what bound, depending just on n, can we place on the size of a field required to represent M? To that end, let Mn be the set of all representable matroids on n elements. For a matroid M , let c(M) be the least positive characteristic of a field over which M is representable. For each positive integer n, define c(n) = max{c(M) : M ∈ Mn}. Department of Pure Mathematics, University of Waterloo, Canada. jpbell@uwaterloo.ca School of Mathematics and Statistics, Victoria University of Wellington, New Zealand. Current affiliation: Dept. of Mathematics, Douglas College, Canada. funkd@douglascollege.ca School of Mathematics and Statistics, Victoria University of Wellington, New Zealand. byoungdu.kim@vuw.ac.nz, dillon.mayhew@vuw.ac.nz

Given a 3-connected biased graph Ω with a balancing vertex, and with frame matroid F(Ω) nongraphic and 3-connected, we determine all biased graphs Ω′ with F(Ω′)=F(Ω). As a consequence, we show that if M is a 4-connected nongraphic frame matroid represented by a biased graph Ω having a balancing vertex, then Ω essentially uniquely represents M. More precisely, all biased graphs representing M are obtained from Ω by replacing a subset of the edges incident to its unique balancing vertex with unbalanced loops.

Let M be a representable matroid on n elements. We give bounds, in terms of n, on the least positive characteristic and smallest field over which M is representable. Our starting point is given by the following two theorems of Rado [5]. Theorem 1 (Rado, 1957). Let M be a matroid representable over a field K. Then M is representable over a simple algebraic extension of the prime field of K. Theorem 2 (Rado, 1957). Let K be an extension field of Q of degree N , and let M be a matroid representable over K. Then there is a positive integer c such that given any prime p > c there is a positive integer k = k(p) ≤ N such that M is representable over GF (pk). For infinitely many p, k(p) = 1. Together, these two theorems say that if a matroid is linearly representable, then it is representable over a finite field. We ask, given a representable matroid on n elements, how large must such a field be? That is, given an n-element representable matroid M , what bound, depending just on n, can we place on the size of a field required to represent M? To that end, let Mn be the set of all representable matroids on n elements. For a matroid M , let c(M) be the least positive characteristic of a field over which M is representable. For each positive integer n, define c(n) = max{c(M) : M ∈ Mn}. Let f(M) be the order of the smallest field over which M is representable. For each positive integer n, define f(n) = max{f(M) : M ∈ Mn}. By Rado’s Theorems 1 and 2 above, c(n) exists and f(n) is finite for all n. Note that c(n) ≤ f(n) for all n, and that, since adding a loop to an n-element matroid yields a matroid on n+1 elements representable over exactly the same fields, c and f are non-decreasing. A result of Brylawski [1] provides a lower bound for c (and thus for f ; see Section 4). We ask for upper bounds on c(n) and f(n). For matroids on at most 8 elements, Table 1 summarises the data (the fact that f(8) = 11 is courtesy G. Royle [personal communication]. We obtain the following bounds. Date: March 2, 2018. Supported by a Rutherford Discovery Fellowship. 1 2 BELL, FUNK, KIM, AND MAYHEW n c(n) f(n) 1 2 2 2 2 2 3 2 2 4 2 3 5 2 4 6 2 5 7 3 7 8 ? 11 Table 1 Theorem 3. For all positive integers n, log2 log2 c(n) ≤ n and log2 log2 log2 f(n) ≤ n. The following fact falls out of the proof of Theorem 3. Theorem 4. Let M be an n-element matroid representable over a field of characteristic 0, and let p be a prime satisfying log2 log2 log2 p > n . Then M is representable over GF(p). We consider the cases of representability over only positive characteristic (Theorem 2.1) and representability over characteristic 0 (Theorem 3.1) separately. Theorem 3 then follows immediately from these results. By Table 1, we may assume throughout the rest of the paper that n > 7. 1. Bounding the degree of a field extension Our first step is to prove an effective version of Rado’s Theorem 1: Theorem 1.1. Let M be a matroid on n elements representable over a field K. Then M is representable over a simple algebraic extension of the prime field of K of degree at most 22 2n 2 . 1.1. A system of polynomials arising from a matroid. Our approach is a standard one in studies of representability of matroids over fields. We assign to an n-element, rank-r matroid M an r × n matrix A whose entries are indeterminates x1, . . . , xt, where t = rn. Each element of the matroid is represented by a column of the matrix. From this matrix we obtain a system of polynomial equations in Z[x1, . . . , xt] as follows. For each r-element subset X of the ground set of M , there is a corresponding r× r submatrix of A whose columns are those representing the elements in X. Setting the determinants of r × r submatrices corresponding to dependent sets to zero, and demanding that the determinants of those r × r submatrices that correspond to bases be nonzero, yields a system of polynomials. The latter conditions may be expressed by multiplying each polynomial fi obtained

Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let M be one of these three classes, and let r be a positive integer. We show that M has only a finite number of excluded minors of rank r.

We give upper and lower bounds on the number of delta-matroids, and on the number of even delta-matroids.

We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list E of 18 such matroids, and show that if N is such an excluded minor, then either N∈E or N is a 2-sum of U2,4 and a 3-connected non-binary frame matroid.

Let $M$ be a frame matroid or a lifted-graphic matroid and let $(G,\mathcal{B})$ be a biased graph representing $M$. Given a field $\mathbb{F}$, a canonical $\mathbb{F}$-representation of $M$ particular to $(G,\mathcal{B})$ is a matrix $A$ arising from a gain function over the multiplicative or additive group of $\mathbb{F}$ that realizes $(G,\mathcal{B})$. First, for a biased graph $(G,\mathcal{B})$ that is properly unbalanced, loopless, and vertically 2-connected, we show that two canonical $\mathbb{F}$-representations particular to $(G,\mathcal{B})$ are projectively equivalent iff their associated gain functions are switching equivalent. Second, when $M$ has sufficient connectivity, we show that every $\mathbb{F}$-representation of $M$ is projectively equivalent to a canonical $\mathbb{F}$-representation; furthermore, when $(G,\mathcal{B})$ is properly unbalanced, the canonical representation is particular to and unique with respect to $(G,\mathcal{B})$.

A frame matroid M is graphic if there is a graph G with cycle matroid isomorphic to M. In general, if there is one such graph, there will be many. Zaslavsky has shown that frame matroids are precisely those having a representation as a biased graph; this class includes graphic matroids, bicircular matroids, and Dowling geometries. Whitney characterized which graphs have isomorphic cycle matroids, and Matthews characterised which graphs have isomorphic graphic bicircular matroids. In this paper, we give a characterization of which biased graphs give rise to isomorphic graphic frame matroids.

A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs on $G$ arise from orienting $G$ and then labelling the edges of $G$ with elements of a group $\Gamma$. In this case, we may define a biased graph by declaring a cycle to be balanced if the product of the labels on its edges is the identity, with the convention that we take the inverse value for an edge traversed backwards. Our first result gives a natural topological characterisation of biased graphs arising from group-labellings. In the second part of this article, we use this theorem to construct some exceptional biased graphs. Notably, we prove that for every $m \ge 3$ and $\ell$ there exists a minor minimal not group labellable biased graph on $m$ vertices where every pair of vertices is joined by at least $\ell$ edges. Finally, we show that these results extend to give infinite families of excluded minors for certain families of frame and lift matroids.