Papers

We show that the isomorphism relation for countable Steiner triple systems is Borel complete, that is, the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Steiner triple systems. To prove it, we construct a faithful Borel reduction from countable graphs to countable Steiner triple systems, that is, a Borel assignment $\theta$ that associates every countable graph $G$ with a countable Steiner triple system $\theta(G)$ so that $G\cong G'$ if and only if $\theta(G)\cong\theta(G')$. Moreover, $\theta$ preserves automorphisms which means that $\mathrm{Aut}(G)\cong\mathrm{Aut}(\theta(G))$.

We prove two surprising results about phylogenetic networks. First, we show that the structure of tree-child networks with $n$ leaves and $k$ reticulation nodes undergoes a sharp phase transition at $n^{1/3}$: if $k=o(n^{1/3})$, then a random tree-child network is almost surely a semi-simplex tree-child network, whereas if $k/n^{1/3}\rightarrow\infty$ and $k=o(n^{1/2})$, it is almost surely not. Second, we show that this result implies that the asymptotic counting formula for galled networks with $n$ leaves and a fixed number $k$ of reticulation nodes remains valid in the range $k=o(n^{1/3})$, but not beyond. This is in strong contrast to recently established results for the asymptotic counting formulas for tree-child and normal networks with $n$ leaves and $k$ reticulation nodes, which are valid in the (optimal) range $k=o(n^{1/2})$.

The Tutte polynomial, through its rank-generating formula, provides a unified framework for describing various combinatorial structures of graphs and matroids, and serves as an important polynomial invariant connecting graph theory, matroid theory, and their related applications. Let $G=(V(G),E(G),r(G))$ be a simple rooted graph, where $V(G)$ is the vertex set, $E(G)$ is the edge set, and $r(G)$ is the root. Let $\Gamma(G)$ be the greedoid induced by the rooted graph $G$, and let its greedoid Tutte polynomial be denoted by $T(\Gamma(G);x,y)$. Let $r=r(G)$, $L_r(G)=\{v\in V(G)\setminus\{r\}: rv\in E(G),\ d_G(v)=1\}$, and $\ell_r(G)=|L_r(G)|$. Gordon and McMahon proposed the following conjecture: for rooted graphs, $\operatorname{ord}_{x}T(\Gamma(G);x,y)=\ell_r(G)$, where $\operatorname{ord}_{x}T(\Gamma(G);x,y)=\max\{a\in\mathbb Z_{\geq0}:x^a\mid T(\Gamma(G);x,y)\}$, that is, the highest power of $x$ dividing $T(\Gamma(G);x,y)$ is equal to the number of leaf vertices adjacent to the root. In this paper, we proved that this conjecture holds for all simple rooted graphs.

The van der Waerden number $w(k)$ is the smallest positive integer $N$ such that every two-coloring of $\{1,2,\ldots,N\}$ contains a monochromatic $k$-term arithmetic progression. We prove that $w(k) \geq (1-o(1))k2^{k-1}$ holds for all positive integers $k$. This verifies a conjecture of Erdős. In 1968, Berlekamp proved the same result when $k-1$ is prime. The coloring for general $k$ can be viewed as a product of Berlekamp's colorings for various primes. It was found by ChatGPT 5.6 Sol Pro.

Resistance curvature, introduced by Devriendt and Lambiotte, is a novel discrete curvature notion defined through effective resistance. A graph is called resistance nonnegative if there exists a choice of positive edge weights for which the resistance curvature is nonnegative at every vertex. This property has a notable combinatorial interpretation in terms of random spanning trees: a graph is resistance nonnegative if and only if it admits a distribution on its spanning trees under which every vertex has expected degree at most two. The resistance nonnegativity can also be characterized by the tree double matching polytope. These characterizations reveal strong connections among resistance curvature, effective resistance, spanning tree distributions, matching theory, and polyhedral combinatorics. Based on the sign of the curvature, Devriendt introduced the classes of resistance nonnegative (RN), resistance positive (RP), and strictly resistance nonnegative (SRN) graphs, and posed several questions concerning their recognition, polyhedral structure, and structural properties. In this paper, we first answer Devriendt's question on the computational complexity of recognizing RN, RP, and SRN graphs by proving that all three classes can be recognized in polynomial time. We then address his question concerning the tree double matching polytope $\Theta(G)$. Further, we characterize the vertices of $\Theta(G)$ in terms of full-rank systems of tight constraints. Whenever $\Theta(G)\neq\emptyset$, we also determine the least positive integer $k_G$ such that $k_G\Theta(G)$ is a lattice polytope. Finally, for every finite Cartesian product of paths, we explicitly construct an average point satisfying the condition for resistance nonnegativity, thereby obtaining that such graphs are RN. We further characterize the classes of such Cartesian product graphs that are RP or SRN.

The trace of the average mixing matrix of a quantum walk measures the "laziness" of the walk: the higher the trace, the more likely that the walker returns home in the long run. In this paper, we develop tools to study this graph invariant arising from Laplacian quantum walks. It is known that the complete graph $K_n$ is the laziest connected graph on $n$ vertices. Using our machinery, we show that the star $S_n$ is the second laziest connected graph on $n$ vertices (and hence the laziest tree on $n$ vertices), the complete multipartite graph $K_{n-2,1,1}$ is the third laziest connected graph on $n$ vertices, and the double star $DS(n-3,1)$ is the second laziest tree on $n$ vertices. We also show that on the same number of vertices, more unbalanced double stars are lazier.

Let \(H\) be a finite hypergraph with rank $r$ and minimum degree \(d\). In the Maker–Breaker degree game, Maker and Breaker alternately claim previously unclaimed hyperedges of \(H\), with Maker moving first, and Breaker seeks to maximize the minimum degree $\delta(H_{\mathrm B})$ of his spanning subhypergraph $H_{\mathrm B}$. We prove that, for every fixed \(r\ge2\) and all sufficiently large \(d\), Breaker has a deterministic strategy satisfying \[ \delta(H_{\mathrm B})\ge \frac{d}{r+1}-\sqrt{d\log d}. \] For graphs, this improves the classical universal lower bound \(d/4\) to \(d/3-\sqrt{d\log d}\). The proof introduces a virtual balancing process, encodes local imbalance by a multiplicative risk, and keeps the resulting risk vector in the Shearer region throughout the game.

Let \(d\ge2\) be fixed and let \(G_n\) be finite abelian groups of odd orders \(N_n\to\infty\). We determine the centered diameter-\(d\) critical window for the standard random Cayley graph in which each nonzero group element is selected independently. Writing \(M_n=(N_n-1)/2\), we prove that the normalized first distance-\(d\) coverage times satisfy \sum_{[x]\in(G_n\setminus\{0\})/\{\pm1\}} \delta_{\frac{N_n^{d-1}}{d!}\tau_{n,[x]}^d-\log M_n} \xrightarrow{d} \PPP(e^{-z} $\,dz). Consequently, the number of antipodal defects in the critical window converges in total variation to a Poisson law, the diameter transition has the Gumbel profile \(e^{-e^{-c}}\), and the diameter hitting time has Gumbel fluctuations. In the original generator-density parametrization this yields the sharp fixed-\(d\) threshold constant \(d!/2^d\) throughout the odd-order abelian class. For \(d=2\), we additionally obtain an exact path–cycle decomposition of the target representation graphs.

We consider systems of homogeneous multivariate polynomials equivariant under the complex reflection group $G(r,n) = (\mathbb{Z}/r\mathbb{Z})^n \rtimes S_n$. Using divided differences indexed by partitions of $n$, we establish a decomposition formula expressing the resultant of such a system as a product of resultants of smaller, partition-indexed subsystems. Combining this with the classical resultant–discriminant relation, we show that the discriminant of a $G(r,n)$-invariant homogeneous polynomial splits explicitly into a product of resultants of smaller subsystems, considerably easier to compute; we illustrate both decompositions with worked examples.

We study how spherical $2$-designs arising from finite group orbits can be completed to spherical $4$-designs by adjoining further orbits, allowing weights in the general theory. For an irreducible real orthogonal $G$-module $W$ with $\mathbb D=\operatorname{End}_G(W)\in\{\mathbb R,\mathbb C,\mathbb H\}$, we consider the multiplicity-two representation $W\oplus W$ and retain the failure of the $2$-design equation $M^*M=\frac12 I_2$ as a quadratic defect. When the invariant quartics are determined by the Hermitian Gram matrix, the fourth-moment problem reduces to a mean and covariance condition on these defects. This yields a sharp lower bound for the total weight of the correction orbits; at equality, their normalized defects form a weighted spherical $2$-design in the associated defect space. The quartic condition holds for the multiqubit Clifford groups in every dimension $r\geq 1$, giving an unbounded-dimensional family with a fixed three-dimensional defect space; among equality cases using the minimum number of correction orbits, the defect geometry is always a regular tetrahedron. As a complementary unweighted example, we construct a $378$-point $W(E_6)$-invariant spherical $4$-design in $S^{11}$ and prove that it is sharp among unweighted invariant completions containing a spherical $2$-design orbit.

In 1984, Roland Häggkvist posed the problem of constructing Hamiltonian graphs of order $n$ with large minimum degree and no $(n-2)$-cycle. He remarked that he did not know of such a graph with minimum degree at least three. We solve this problem by proving the following two results. (1) For every integer $d\ge 3$ and every integer $n\ge 15d-14,$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-2)$-cycle. (2) For every integer $d\ge 3,$ every positive integer $k,$ and every integer $n\ge (k+1)[(d-1)(k+3)+1],$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-s)$-cycle for any $s\in\{1,2,\dots,k\}.$ The proofs are constructive. We also pose several open problems.

An $st$-orientation of an undirected graph $G$ is an acyclic digraph with a single source $s$ and a single sink $t$ that can be obtained from $G$ by assigning a direction to each edge. The classical problem of deciding if an undirected graph $G$ has an $st$-orientation can be solved efficiently. On the other hand, deciding if an $st$-orientation of $G$ exists that does not have any transitive arc is NP-complete, even if each vertex of $G$ has degree at most four. Here we show that this last decision problem remains NP-complete if $G$ is cubic, which settles an open question by Binucci et al. (2025). We obtain NP-completeness for two variants of the problem: (i) $s$ and $t$ are fixed and given as part of the input and (ii) $s$ and $t$ can be chosen freely. We then use these results to investigate the computational complexity of a problem that arises in computational evolution. Specifically, we show that the problem of deciding if an unrooted binary phylogenetic network has an orientation as a rooted binary phylogenetic network without any shortcuts (the analog of a transitive arcs in phylogenetics) is NP-complete. Our results connect the two (mostly) distinct research areas of orienting undirected graphs and orienting unrooted phylogenetic networks.

In this article, we survey some fundamental results and recent advances in the design and analysis of Boolean bent functions from the perspective of their relation to the completed Maiorana-McFarland class.

Baker, Huh, Kummer, and Lorscheid define the triangular-hyperfield threshold q(n)=q(U_{2,n}) and conjecture exact values for all n. Using their identity q(n)=P(n-1), where P(m) is the universal negative-type exponent of m-point Ptolemaic metrics, we prove q(6)=\log_2(9/4) and q(7)=1. For five points, we separate zero-sum coefficient vectors by sign pattern. A sharp two-summand inequality proves the inequality for the 1+4 pattern. For the 2+3 pattern, a sharp four-point partial-correlation bound and an exact copositivity identity prove the required inequality. For six points, we construct an involutive coefficient transport under metric inversion that preserves the negative-type quadratic form. Applying the transport at an index with positive local contribution converts a hypothetical 3+3 counterexample into a 2+4 counterexample. Complete split graph metrics attain both bounds.

We extend, and fully implement, in Maple, Jim Propp's charming way of estimating Pi via tossing a fair coin, and compute many other constants, even going beyond the far more general set-up of Bruss and Paindaveine.

For an integer $q\ge2$, a matroid is $q$-entropic if its rank function, multiplied by $\log q$, is the joint-entropy function of random variables on a $q$-element alphabet. We prove that a matroid is $4$-entropic if and only if it is representable over $\F_4$. The corresponding statements for alphabet sizes two and three were known. The proof combines minor closure and the excluded-minor characterization of quaternary matroids with structural properties of quasigroups of order four. Thus arbitrary four-symbol partition representations yield no matroids beyond the quaternary ones. As an application, every access structure admitting an ideal perfect scheme with a uniform four-symbol secret and four-symbol active shares also admits an ideal $\F_4$-linear scheme.

Let $G$ be a finite simple co-chordal graph with at least one edge, and let $I(G)$ be its edge ideal. Over an arbitrary field, we prove that the symbolic powers of $I(G)$ satisfy $\operatorname{reg} I(G)^{(s)}=2s$ for every $s\ge1$. Thus every symbolic power of $I(G)$ has a degree resolution. Using Takayama's formula and clique trees, we reduce the regularity problem to a topological one. Finite convex geometry then shows that nontrivial homology implies a suitable set of leaves, and a weighted counting argument gives the required regularity formula. Finally, we show that symbolic powers of co-chordal edge ideals are not necessarily componentwise linear.

A sequence $a_1,\ldots,a_n$ of integers is nearly consecutive if $a_{i+1}-a_i \in \{1,2\}$ for $1 \leq i \leq n-1$. We prove that there are nearly consecutive sequences of length $\Omega(2^k/k^2)$ that contain no $k$-term arithmetic progression. This improves on the previous best known bound of Alon and Zaks from 1998. We also prove a generalization for sequences with bounded gaps.

We determine the sharp exponential growth rate of the minimum number of abelian subgroups required to cover a group with bounded pairwise noncommutativity. Let $\omega(G)$ denote the largest size of a pairwise noncommuting subset of a group $G$, let $a(G)$ be the least size of an abelian cover, and define $h(n)=\sup\{a(G):\omega(G)\le n\}$. We prove the quantitative estimate $\log_2 h(n)=n/2+O(\sqrt{n}\,(\log(n+2))^3)$, and hence $h(n)^{1/n}\to\sqrt{2}$. Extraspecial $2$-groups give the matching lower bound. For the upper bound, we reduce to finite groups by isoclinism, analyze finite $p$-groups through a central series of the derived subgroup and alternating commutator forms, control interactions between central factors, and then pass through Sylow decomposition and the Fitting subgroup at polynomial cost. The argument also determines the same sharp exponential rate for the least possible index of an abelian subgroup and shows that asymptotic extremality is concentrated in $2$-groups. This result resolves Erdős Problem #117 at the level of its sharp exponential asymptotics.

We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial $p$-group of order $p^{2n+1}$ with a conjugacy-closed connection set is a Cayley graph of $Z_p^{2n+1}$. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, $V_{8n}$ and extraspecial $2$-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power $q\ge 5$, the $SL(2,q)$ graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most $30$ vertices.

An efficient $k$-limited dominating broadcast, or $k$-ELDB, is a $k$-limited broadcast in which every vertex is dominated exactly once. This notion brings together efficient domination and limited broadcast domination in a common framework. For a graph $G$, we write $mcr(G)$ for the smallest integer $k$ for which $G$ admits a $k$-ELDB. For an admissible value $k\ge mcr(G)$, we denote by $\gamma_{ebk}(G)$ the minimum cost of a $k$-ELDB on $G$, and is called the $k$-efficient broadcast domination number of $G$. In this paper, we study these parameters from an algorithmic perspective with complexity analysis. We develop a dynamic programming algorithm for trees which, for fixed $k$, computes $\gamma_{ebk}(T)$ and thereby obtains a polynomial-time procedure for determining $mcr(T) $. In contrast, we prove that, for every fixed integer $k\ge 1$, deciding whether a graph admits a $k$-ELDB is NP-complete for arbitrary graphs. These results place efficient limited broadcast domination in a natural complexity framework, with trees forming a tractable class and arbitrary graphs remaining computationally hard.

The Kahn–Lovász theorem gives a sharp upper bound on the number of perfect matchings in a graph in terms of its degree sequence, extending the classical Brégman–Minc inequality for bipartite graphs. In this paper, we establish an asymptotically sharp extension of the Kahn–Lovász theorem to $F$-factors for every Hamiltonian graph $F$. As a consequence, we asymptotically determine the maximum number of $F$-factors in an $n$-vertex $m$-edge graph, yielding an $F$-factor analogue of Kruskal–Katona-type theorems. We also prove a multigraph analogue of the Kahn–Lovász theorem. Combining this with our results for Hamiltonian graphs, we obtain an asymptotically sharp Kruskal–Katona-type bound for a further class of connected graphs $F$, including those containing two vertex-disjoint cycles of equal length whose union spans $V(F)$.

The growth rate of Av(1324) is the last unknown Stanley-Wilf limit of a length-four pattern. Since 2020 the best rigorous lower bound has been the 10.271012 of Bevan, Brignall, Elvey Price and Pantone; we raise it to 10.412263. Their construction lays 1324-avoiders along an infinite staircase of two-cell blocks alternating with single connecting cells, and forbids a 1324 by a local condition on how the points of a block interleave with the components of the cell beside it. Certain points of a block, its leaves, may be exempted from that condition. They exempt them on one axis only, and report that they could not count the possibilities when both are exempted. This paper exempts them on both axes and counts the result. The second relaxation is their own argument read in the other axis, and the Harris correlation inequality bounds the joint count below by the product of the two counts taken separately, at the cost of one factor of the component generating function.

Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $\sigma^{-1}\tau$ has a fixed point for all $\sigma,\tau\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-sized intersecting families of permutations in $\mathrm{Sym}(n,k)$. In the regime $k\leq n^{0.25}$, we show that every maximum-sized intersecting family is a star, meaning it consists of all permutations in $\mathrm{Sym}(n,k)$ that agree at a given point in $[n]$. We establish this result by proving a stronger stability result that bounds the maximum possible size of a non-centred intersecting family. Specifically, in the regime $k\leq n^{0.25}$, the size of any non-centred intersecting family is at most $\left(2/3+o(1)\right)$ times the maximum possible size of a star. In the tighter polylogarithmic regime $k\leq (\ln n)^{d}$, we improve this bound to $\left(1-1/e+o(1)\right)$ times the maximum possible size of a star; we show that this bound is asymptotically sharp. Thus, we establish both an Erdős–Ko–Rado theorem and its corresponding stability version for $\mathrm{Sym}(n,k)$.

Frankston, Kahn and Narayanan proved that every regular increasing $3$-wise intersecting family of subsets of $[n]$ has cardinality $o(2^n)$ using Friedgut's junta theorem. We give a short quantitative proof using elementary tools from the analysis of Boolean functions and entropy. More precisely, if $\mathcal{A}\subseteq\mathcal{P}_n$ is a nonempty $3$-wise intersecting family that is both regular and increasing, then $$ \log\frac{2^n}{|\mathcal{A}|}\ge \frac{n}{2}\left(\frac{|\mathcal{A}|}{2^n-|\mathcal{A}|}\right)^2, $$ and consequently $|\mathcal{A}|\le 2^n\sqrt{W(n)/n}$, where $W$ is the principal Lambert function defined by $W(x)e^{W(x)}=x$ for $x\ge0$. We also give a purely Fourier-analytic proof of the weaker estimate $$ |\mathcal{A}|\le \frac{2^n}{1+n^{1/3}}. $$

The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin–Shapiro, and Thue–Morse words, words coming from coding a rotation by intervals, and bracket words.

Sublinear expansion is weak enough to be extracted from arbitrary graphs while retaining nearly all of their average degree, yet it has proved strong enough to force global structures in many sparse extremal problems. Letzter, Methuku and Sudakov [JLMS 2026] proved the existence of nearly Hamilton cycles in sufficiently dense regular sublinear expanders. Montgomery [ICM 2026] subsequently conjectured that, every $d$-regular sublinear expander with $d$ sufficiently large (but constant) is Hamiltonian. We disprove this conjecture in a strong form by constructing $n$-vertex $d$-regular sublinear expanders with $d=\left(\frac12+o(1)\right)\log^2 n$, which can forbid any cycle covering an arbitrarily small given positive constant portion of its vertices. The construction blows up one side of a biregular Ramanujan graph into almost-complete blocks and keeps the other side as a sparse vertex separator. We also prove a similar statement for a closely related notion of edge expanders. For every sufficiently small $\gamma>0$, there is an infinite family of $n$-vertex $d$-regular $\gamma$-edge-expanders with $d=\Theta(\gamma^{-1})$ and circumference $O(\gamma n)$, matching the standard lower bound $\Omega(\gamma n)$. The construction also comes from an expanding regular core such that each vertex has an almost-complete graph attached to it. The core guarantees edge expansion, while the single-vertex attachments confine every cycle.

Random access is a central requirement in DNA-based storage systems: one would like to recover selected information symbols without sequencing the whole encoded object. A recent combinatorial model associates to a generator matrix $G\in F_q^{k\times n}$ the random variable $\tau_i(G)$, measuring the number of sampled columns needed to recover the information vector $e_i$. We study the cardinality-based extremal and finite-geometric aspects of simultaneous multi-symbol recovery. For a nonempty set $I\subseteq[k]$, let $\tau_I(G)$ denote the number of random column samples needed until all vectors $e_i$, $i\in I$, lie in the span of the observed columns. This variable interpolates between the singleton random access problem and the full-recovery problem underlying coverage depth. For each $m$, we introduce uniform worst-case and average parameters over all requested sets $I$ with $|I|=m$. Using the known subset-counting formula for $E[\tau_I(G)]$, we establish general upper and lower bounds for these parameters. In particular, the lower bounds are expressed through order statistics of the singleton recovery variables and specialize to the known singleton bounds when $m=1$. For systematic MDS encoders, we record an equivalent form of the known multi-symbol expectation formula and derive monotonicity and asymptotic consequences. For simplex encoders in arbitrary dimension, we obtain closed formulae in terms of Gaussian binomial coefficients; the full-recovery endpoint agrees with the known coverage-depth formula for simplex codes. Finally, in dimension three we study balanced quasi-arcs and compare their values with the simplex and MDS benchmarks.

Let $Q^\varepsilon(3,q)$, where $\varepsilon\in\{+,-\}$ and $q>3$ is odd, be a non-degenerate hyperbolic or elliptic quadric of $PG(3,q)$. Fix one of the two quadratic classes of anisotropic points. Since the line joining two distinct points of this class is tangent, secant, or external to the quadric, one obtains a $3$-class association scheme. We classify all non-trivial unions of its relations which define Deza graphs. In addition to the previously known tangency family, exactly four exceptional strictly Deza graphs occur, with parameters $(360,135,54,45)$, $(369,108,36,27)$, $(65,34,18,15)$ and $(168,111,75,70)$. We determine their spectra and Deza children and give geometric or group-theoretic descriptions of all four exceptional graphs.

We study finite, connected, simple bipartite graphs in a grid model, in which a graph is drawn as a rectangular array and its structure is read off from empty subrectangles, called holes. In this model we attach to every brick a numerical invariant, its characteristic m, the difference between the number of rows and the largest proper independent set. A brick is excessive if m > 0. Our main results concern this invariant. We determine the characteristic of a disconnected excessive brick from those of its components, showing that m = min_i min{m_i, imb(W_i)} while the imbalance is additive; and we prove that an m-excessive brick is m-extendable, that is, every matching of size m extends to a maximum matching. Since Plummer's notion of n-extendability is defined only for graphs carrying a perfect matching, and our proof nowhere uses balance, the characteristic extends that notion canonically to unbalanced bipartite graphs. Using the characteristic we partition bipartite graphs into eleven structural classes. The underlying decomposition into atomic blocks is the classical decomposition into elementary components, and the description of the maximum proper independent sets by ideals of the block poset is likewise classical; the paper states precisely which results are classical and are not claimed here. What the grid model adds is a single geometric framework in which holes, characteristics and the block triangular form are read off from one picture.