Papers

We prove RE-completeness of the quantum homomorphism problem parameterised by families of graphs derived from the classic metric association schemes. These include Kneser graphs, $q$-Kneser graphs, and the complements of Johnson, Grassmann, and Hamming graphs. Our proof develops a spectral method for establishing non-contextuality of quantum polymorphisms. It combines an equality analysis of Roberson's bound on the projective packing number in terms of Schrijver's theta with a structural argument inspired by Erdős-Ko-Rado theory.

Framing lattices arise from the dual graphs of framed (or DKK) triangulations of flow polytopes and provide a common framework encompassing classical lattices such as the Boolean, Tamari, and weak-order lattices, as well as $\tau$-tilting posets of certain gentle algebras. In this paper, we introduce the framingtope, a polytopal complex that provides a geometric counterpart to a framing lattice: its edge graph is the Hasse diagram of the framing lattice. We prove that the framingtope admits three equivalent descriptions, in terms of interior faces of the framed triangulation, sets of pairwise coherent routes covering the graph, and pure intervals of the framing lattice. We further construct a tropical realization of the framingtope as the bounded-cell complex of an arrangement of tropical hypersurfaces associated with an admissible height function. This construction yields explicit vertex coordinates for broad classes of framed graphs, including plane framed graphs and multioruga graphs. In the multioruga case, these coordinates give tropical realizations of weak orders on multipermutations and, in the ordinary oruga case, recover the classical permutahedron.

Let $(d_i(n))_{i=0}^n$ be the Boros–Moll coefficient sequence. We prove that, for every integer $n\ge1$, the polynomial \[ M_n(x)=\sum_{i=0}^n \bigl(d_i(n)^2-d_{i-1}(n)d_{i+1}(n)\bigr)x^i \] has only simple negative zeros, which strictly interlace those of the Narayana polynomial of the same degree. This proves a conjecture of Chen, Yang, and Zhang and, by Brändén's preservation theorem, settles the infinite log-concavity conjecture of Boros and Moll. The proof uses an expansion of the reversed and normalized form of $M_n(x)$ in derivatives of the Narayana polynomial, together with estimates for the weights and partial sums of the normalized derivatives.

We investigate the behavior of arboricity under two fundamental graph operations, namely wedges and joins, proving an exact formula for wedges and establishing general upper and lower bounds for joins. Using the characterization of the simplicial geometric category of connected graphs in terms of arboricity, we derive a wedge formula for simplicial geometric category and obtain corresponding estimates for graph joins. Finally, we illustrate these results through explicit computations for several classes of graphs by constructing forest decompositions and the associated covers by strongly collapsible subcomplexes.

We prove that the height-level path ensemble of a uniform alternating sign matrix converges to the Airy line ensemble near every interior point of the north-west arctic arc. Under the standard bijection, this is equivalently a limit theorem for the osculating paths of the domain-wall six-vertex model at the ice point. The proof has two main inputs. The integrable input is a new Pfaffian formula for rectangular frozen-corner probabilities, from which we obtain one-point convergence of the top path to the GUE Tracy–Widom distribution. The probabilistic input is an exact resampling law and accompanying bridge estimates for a strictly ordered midpoint encoding of the paths; these yield tightness and the Brownian Gibbs property for every subsequential limit. Aggarwal and Huang's strong characterization then identifies each such limit as the Airy line ensemble.

Let $f(n,t)$ be the minimum number of triangles in a tripartite graph with $n$ vertices in each part and minimum degree at least $n+t$. In 1975, Bollobás, Erdős and Szemerédi proved that $f(n,1)=\min\{4,n\}$. They further remarked that it is “very likely” that $f(n,t)\ge4t^3$ for $n\ge5t$. They also proved that $f(n,t) \ge t^3$ for all integers $n \ge t \ge 1$. We construct graphs showing that, for all integers $t\ge1$ and $n\ge3t+2\lceil(1+\sqrt5)t/2\rceil$, \[ f(n,t)\le(1+\sqrt5)t^3+\left(1+\frac1{\sqrt5}\right)t^2. \] Here $1+\sqrt5\approx3.236<4$, and the displayed upper bound is strictly less than $4t^3$ for every $t\ge2$, disproving their proposed bound. We also improve their lower bound $t^3$ by showing that $f(n,t)\ge\frac{12}{5}t^3$ for all integers $t\ge2$ and $n\ge18t^6$.

We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers–Ramanujan and Andrews–Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1<a<b.$ For a prime power $q$, let $\mathcal{NC}_n^{a,b}(\mathbb F_q)$ denote the set of pairs of commuting nilpotent $n\times n$ matrices $(A,B)$ over $\mathbb F_q$ satisfying $A^a=B^b$. We establish the threefold equality between their normalized counts, the HJO $q$-series $Z_{a,b}$, and the explicit infinite product $P_{a,b}$: \[ \underbrace{\vphantom{\Bigg|} \prod_{m\geq1}(1-q^{-m}) \Biggl(\sum_{n=0}^{\infty} \frac{\lvert\mathcal{NC}_n^{a,b}(\mathbb F_q)\rvert} {\lvert\operatorname{GL}_n(\mathbb F_q)\rvert}\Biggr) }_{\text{point count}} = \underbrace{\vphantom{\Bigg|}Z_{a,b}(q^{-1}) }_{\text{\(q\)-series}} = \underbrace{\vphantom{\Bigg|}P_{a,b}(q^{-1}) }_{\text{infinite product}}. \] Our main result is a stronger finite identity: the rank $N$ HJO sum equals $(q;q)_N$ times the generating function for balanced cylindric partitions with entries bounded by $N$. Taking $N\to\infty$ yields the HJO conjecture. The proof combines the compositional rational shuffle theorem of Bergeron–Garsia–Leven–Xin and Mellit with a multiplicativity theorem for slope operators and a determinantal model for bounded cylindric partitions, linked by a common $q$-difference equation. The finite identity and the HJO conjecture have been formalized unconditionally in Lean by AxiomProver.

We show that the uniform probability measure on copies of a nonempty graph $H$ in $K_n$ is $Cq_H\log(2e(H))$-spread, where $q_H$ is its graphic expectation threshold. Consequently, the fractional expectation threshold of $H$ is at most $Cq_H\log(2e(H))$. We remove the logarithmic factor for trees and for graphs whose average degree is at least the logarithm of their maximum degree. This proves the “second” Kahn-Kalai conjecture for these two classes, which encompass most of the standard families studied in random graph containment problems.

By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in $U^k(\Phi)$-uniform and Nil-Bohr sets. To be more precise, our first result associates the degree $k$ of a $U^k(\Phi)$-uniform set with the variety of sumsets it contains, solving a conjecture of Kra, Moreira, Richter and Robertson. Restricting to Nil-Bohr sets we show the existence of infinite sumsets with summands in the shifted primes $\mathbb{P}-1$. As a consequence, we show that for any real polynomial $Q(n)$ with leading irrational coefficient of degree $k$, and any natural numbers $\ell_1, \cdots, \ell_k$ there is an infinite set $P\subset \mathbb{P}$ such that \begin{equation*} Q\Big(\sum_{p \in I} p\Big) \in U \pmod 1 \quad \text{ for all } I \subset P , |I| = \ell_1, \ldots, \ell_k. \end{equation*}

We prove a sharp concentration inequality for the spectral norm of sparse random tensors with independent Bernoulli entries. Let $T$ be an order-$k$ tensor of dimension $n\times\cdots\times n$ with independent Bernoulli$(p)$ entries, where $k$ is fixed. For any $c,r>0$, we show that $\|T-\mathbb E T\|\le C_{k,r,c}\sqrt{np}$ with probability at least $1-n^{-r}$ whenever $np\ge c\log n$. We extend this bound to inhomogeneous Bernoulli sampling with deterministic entrywise weights. This removes the logarithmic factor in the work of Zhou and Zhu (2021). The proof follows the Kahn–Szemerédi light–heavy decomposition with a refined estimate on the heavy tuple part. We also obtain a log-free second eigenvalue bound for the random hypergraph model of Friedman and Wigderson (1995).

In this paper, we explore group invariants arising from combinatorial structures associated with finite groups, including the functional graphs of power maps and the well-studied power graphs. These invariants induce equivalence relations (and hence partitions) on the set of isomorphism classes of finite groups, which we classify from the finest to the coarsest. Surprisingly, all but three of these partitions turn out to coincide; for the subclass of nilpotent groups, all but two coincide. Furthermore, we introduce a broad class of nilpotent groups and show that, within this class, all but one of these partitions agree. Our proofs draw on tools and ideas from Combinatorics and Number Theory, while requiring only basic notions from Group Theory. In particular, we construct a general framework that may prove useful in contexts similar to those considered in this paper. Finally, we propose some open questions that emerge from our results.

We prove that no $52$-point subset of the affine plane $\mathbb F_{13}^{2}$ has exactly four special directions, where a direction is special when its thirteen parallel affine lines do not all meet the set in the same number of points. A universal incidence identity reduces the four exceptional line-count functions to a polynomial identity over $\mathbb F_{13}$. Linear independence of the associated binary forms of degree at least three then forces those functions to have degree at most two. Classification of the resulting constant, linear, and quadratic profiles leaves a quadratic-character congruence with no solution. The incidence and polynomial argument supplies the quantifier over all $52$-point subsets and all four-direction sets; the remaining classification of quadratic value tables is finite and exact. Together with Ghidelli's lower bound and the $65$-point construction of Kiss and Somlai, this determines the minimum size of a subset of $\mathbb F_{13}^{2}$ with exactly four special directions: it is $65$. The AI-assisted workflow used OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, Grok 4.6, and OpenAI GPT-6 Astra, together with Codex-controlled Danus.

A Leech tree of order $n$ is a tree with positive integral edge weights whose $n(n-1)/2$ pairwise weighted distances are precisely $1,2,\ldots,n(n-1)/2$. This paper gives a computer-assisted proof that no Leech tree of order $18$ exists. The argument has three layers. First, a development in Lean 4 verifies the structural facts used in the paper. These facts reduce every putative example to one of eight local configurations and justify several necessary conditions. Second, conventional mathematical arguments prove a component-pair whole-block exact-cover condition and the completeness of a recursive search. Third, exhaustive computations close all eight configurations. The computation records exact coverage, source and input hashes, terminal receipts, and checked exact-zero results. The structural layer is kernel-checked, but the search program, its execution, and the certificate checker have not been formalized in Lean. The result is therefore a computer-assisted proof, not an end-to-end Lean proof.

We show that there exists a set $S \subset \mathbb{Z}^2$ containing no four points on a circle or a line such that $|S \cap [n]^2| = \Omega(n)$ as $n \rightarrow \infty$. Since any no-four-on-a-circle set in $[n]^2$ has size $O(n)$, this resolves (up to a constant) a question raised by the current authors and Keevash concerning the density of extensible no-four-on-a-circle constructions. Our construction is based on weighted random sampling from the integer lattice followed by careful deletion.

A coloring of $\R^n$ is proper for the forbidden distance segment $[1,\ell]$ if no two points of the same color are at a distance from $[1,\ell]$; the minimum number of colors is $\chi(\R^n,[1,\ell])$, and $\ell=1$ gives the classical chromatic number $\chi(\R^n)$ of the Nelson–Hadwiger problem. We prove the new upper bounds $\chi(\R^4)\le43$, $\chi(\R^5)\le132$, $\chi(\R^7)\le1029$, $\chi(\R^9)\le7203$, $\chi(\R^{10})\le45619$, improving the previously known $49$, $140$, $1372$, $17253$ and $3^{10}$; in particular, this refutes the conjecture of Arman, Bondarenko, Prymak and Radchenko that $49$ and $140$ are optimal among all lattice colorings of $\R^4$ and $\R^5$. The first four bounds come from explicit rational lattices — an Eisenstein lattice in $\R^4$, a lattice in general position in $\R^5$, and laminations of the Eisenstein colorings $E_6^*/343$ and $E_8/2401$ in $\R^7$ and $\R^9$ — and each is reduced, by one verification protocol, to a finite list of inequalities between explicitly written rational numbers checked in exact arithmetic. The fifth bound is analytic: we prove that for every Eisenstein lattice $\Lambda$ the distance between same-colored cells of $(3+\omega)\Lambda$ equals $\sqrt{7/3}\,\lambda_1(\Lambda)$, which gives the exact widths of all known colorings with $7^{n/2}$ colors, and a product rule $\sum_i1/d_i^2\le1$ for the widths of orthogonal products; together they yield $45619=2401\cdot19$, the first bound in $\R^{10}$ below $3^n$, as well as $\chi(\R^{25})\le4\cdot7^{12}$ and $\chi(\R^{26})\le19\cdot7^{12}$. We also show that no sublattice of $E_8$ of index below $2401$ defines a proper coloring. All code, exact certificates and data are open.

Let $q=3^m$, let $K=\mathbb F_{q^2}$, and let \[\mathcal T=\{x\in K^*:\operatorname{N}_{K/\mathbb F_q}(x)=1\}.\] For both cyclic and constacyclic Gashkov-Sidel'nikov codes, we show that the set of signed parity-check column labels is precisely $\mathcal T$. Consequently, the decoding problem separates into two stages: determining the minimum error weight associated with a syndrome $S$ and constructing an error vector attaining this minimum. We identify the former quantity with the minimum additive length of $S$ with respect to $\mathcal T$ and determine it exactly by the norm and the quadratic character of $\mathbb F_q$. We also determine the complete coset-weight distribution and recover the known covering radius $3$. For the constructive part, we use quadratic-character sums and Weil bounds to construct a coset leader for every syndrome of coset weight three. The resulting procedures give complete maximum-likelihood decoders.

We consider infinite walks in $\mathbb{N}^k$ with standard unit basis vector steps that avoid $t$ collinear points, and show that these walks exist for $(k,t) \in \{(6,3), (4,4), (3,7)\}$. In particular, our construction for $k = 3$ improves the previous bound $189$, obtained by Lidbetter, to $7$. Our results also imply the existence of infinite words over small finite alphabets that are weakly abelian squarefree (resp., weakly abelian cubefree, weakly abelian 6th-power-free).

We classify, for every prime power $q$ and every $e\geqslant2$, the permutation binomials $X^r(X^{q-1}+a)$ over $\mathbb F_{q^e}$. Writing $\ell_j(q)=(q^j-1)/(q-1)$, such a binomial is a permutation if and only if $\gcd(r,q-1)=1$, $(-a)^{\ell_e(q)}\ne1$, and $r\ell_h(q)\equiv1\pmod{\ell_e(q)}$ for some $1\leqslant h<e$ coprime to $e$. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from $(X^{q^h}+aX)\circ X^r$ for a suitable $h$. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family $X^r(X^{d(q-1)}+a)$ in the coprime-index case $\gcd(d,\ell_e(q))=1$. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas' theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-$q$ digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.

We study random linear projections of a finite-field subset for which every fiber has cardinality close to its mean. We bound the mean fiber size needed to ensure that all fibers satisfy a prescribed relative discrepancy, with a prescribed failure probability. For $S\subseteq\mathbb F_q^n$ projected to $\mathbb F_q^b$, one theorem gives three regimes: at fixed discrepancy and failure probability, sufficient mean fiber sizes are $O(q2^b)$ for arbitrary $q$, $O(q^2)$ when $q$ is at least a suitable constant multiple of $b$, and $O_q(b)$ for fixed $q$. The resulting entropy loss over fixed fields is $h-b=\log_q h+O(1)$, where $h=\log_q|S|$ is the input entropy. This matches the order of the binary obstruction of Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos (1999); we give a quantitative random-source refinement over every fixed field. Our proof combines a quotient-and-average counting lemma with the local balanced/unbalanced argument of Dhar and Dvir (arXiv:2204.01665) and Furstenberg estimates of Dhar and Dvir and Kumar and Mon (arXiv:2609.17020). We apply these bounds in the reduction of Ordentlich, Regev, and Weiss (arXiv:2311.04644) to improve their $O(n^3)$ bound for smooth lattice coverings to $O(n^2)$. For each fixed convex body $K\subseteq\mathbb R^n$, a Haar-Siegel random lattice of covolume one has the number of lattice points in every translate of $K$ within a prescribed relative error of $\operatorname{vol}(K)$, with prescribed high probability, once $\operatorname{vol}(K)\ge Cn^2$ and $n$ is sufficiently large. The constant and dimension cutoff depend only on the error and failure probability. Complements of higher-rank Kakeya sets of Kopparty, Lev, Saraf, and Sudan (arXiv:1003.3736) show that no hashing guarantee for arbitrary subsets can yield a smaller order in the same reduction.

Let $S$ be a set of vertices in a connected graph $G$. A vertex outside $S$ is represented by the multiset of its distances to the vertices of $S$. The outer multiset dimension $\operatorname{odim}(G)$ is the minimum cardinality of an $S$ for which these representations distinguish all vertices outside $S$. We determine $\operatorname{odim}(C_s \square C_t)$ for all $s,t\geq 3$, answering a problem of Klavžar, Kuziak, and Yero. The values range from $3$ to $8$. The proof combines a half-turn argument giving a universal four-landmark lower bound when both factors have length at least four, explicit three- and four-landmark constructions for the infinite families, and exact finite enumeration on the remaining strip. The collision classification behind the infinite four-landmark construction is certified by exact quantifier elimination in linear integer arithmetic; source code and all finite upper certificates accompany the paper.

So conjectured that, for a fixed positive integer n, the ordinary adjacency spectrum of an integral circulant graph of order n determines its divisor set. We prove this for graphs of order p to the power a times q, where p and q are primes with p less than q and a is at least one. To handle coincident eigenvalues arising from distinct greatest-common-divisor classes, we use a spectral counting measure. For connected graphs, an exact identity recovers the part of the divisor set consisting of one and q, when present, together with the counting measure for a graph of order p to the power a minus one times q. Strong induction and decomposition into connected components then recover the full divisor set, including the case p equals two and disconnected graphs.

Löfberg and Qi introduced the second order Zarankiewicz number \(z_2\), the recursive-line Zarankiewicz number \(z_{RL}\), and the signed Zarankiewicz number \(z_{SL}\) for doubly simple biquadratic forms. It was shown that \[ z_2(m,n)\ge z_{SL}(m,n)\ge z_{RL}(m,n) \] for all \(m\) and \(n\). However, there was no evidence that there exist particular \(m\) and \(n\) such that \(z_{SL}(m,n)>z_{RL}(m,n)\). The motivation for introducing \(z_{SL}\) was as follows: during the study of the exceptional case \(m=15\), \(n=6\), Löfberg and Qi showed that \[ z_2(15,6)=z_{SL}(15,6)=60, \] but the exact value of \(z_{RL}(15,6)\) was unknown then. In this paper we show that \[ z_{RL}(15,6)=60. \] This eliminates the motivation for introducing \(z_{SL}\). Whether \(z_{SL}(m,n)=z_{RL}(m,n)\) in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number \(z_A\) from the limited augmented Zarankiewicz number \(z_L\) at \(m=n=1893\). We hope that the separation problem for \(z_{SL}\) and \(z_{RL}\) can also be solved. We also present the exact values of \(z_{RL}(m,6)\) for \(6\le m\le 16\).

Switching methods can be seen as certain local transformations that do not alter their basic parameters of a combinatorial structure. Efforts have been devoted in the literature to relate and unify the switching theories for codes and designs, and also for Hadamard matrices and graphs. The combinatorial structures we consider in this paper are graphs and designs. We show an extension of known switching method for constructing 2-designs to divisible designs, and then provide some examples of its application. Moreover, we prove several equivalences between switching methods for graphs and designs, and as a byproduct, we obtain a new switching method to obtain 2-designs.

Bonamy-Knor-Lužar-Pinlou-Škrekovski (2017) define $K_n^t$ to be the complete graph of $n-1$ vertices but with an extra vertex that's adjacent to $t$ vertices of the complete graph part. They propose a stronger conjecture which asserts that if $G$ is a finite simple $2$-connected graph of order $n \ge 10$ not isomorphic to $K_n$, $K_n^2$, nor $K_n^{n-2}$, then the Szeged-Wiener gap of $G$ is $\eta(G) \ge 2n$. We improve upon their work to tighten the bounds on the Szeged-Wiener gap, allowing us to prove this conjecture in the affirmative. Afterwards, we construct graphs attaining equality for each $n \ge 10$ and pose a problem for interested readers to determine a necessary and sufficient condition for equality.

We show that the face lattice of any simplicial polytope is Hamiltonian. This proves a major part of a recent conjecture by \citeauthor{Listingfaces} which states that the face lattice of any polytope is Hamiltonian (SODA26). We use a line shelling in both directions to obtain two different decompositions of the face lattice into disjoint cubes, which allows the application of a theorem by Gregor to obtain the Hamilton cycle.

This paper introduces and formalizes Fractal Hyper-Trees (FHT), a novel class of hypergraphs characterized by deep recursive inclusion hierarchies and controlled relational textures. Moving beyond classical graph-theoretic frameworks, we establish an axiomatic foundation for FHTs and investigate their combinatorial enumeration, revealing explicit connections to OEIS sequence A398931. Furthermore, we analyze the structural symmetry properties via automorphism groups, examine the resulting global hierarchical ultrametric topologies, and discuss asymptotic perspectives, including vertex substitution operators and heuristic parallels with $p$-adic integer rings.

Orthogonal arrays (OAs) are combinatorial configurations with applications in experimental design, error-correcting codes, and quantum information. Local unitary (LU) equivalence provides a natural framework for classifying multipartite entangled states. Using the correspondence between OAs and quantum states, Goyeneche and Życzkowski [Phys. Rev. A, 2014, 90: 022316] posed the problem of determining when OAs are $LU$ equivalent. In this paper, we construct OAs from generator matrices and establish conditions for Fourier-based $LU$ equivalence of OAs and irredundant orthogonal arrays (IrOAs). For prime alphabets, we identify the Fourier partners of the linear OAs considered here with the arrays of their corresponding dual codes. This gives an explicit connection between $LU$ equivalence and coding theory. We also identify families of $LU$ equivalent OAs arising from specific classes of linear codes.

Let $H_{27}=\mathrm{UT}_3(\mathbb{F}_3)$ be the nonabelian group of order $27$ and exponent $3$. We prove that $\mathsf{d}(H_{27}\times C_3^r)=2r+6$ for every integer $r\geq0$. The proof combines an affine coefficient identity in the group algebra of an elementary abelian group with a decomposition of the nonorthogonality graph of $\mathbb{F}_3^2$ into eight edge-disjoint zero-sum triangles. It is uniform in $r$, does not use the value of the small Davenport constant for a smaller nonabelian group, and requires no computational enumeration.

Alon and Mubayi asked whether, for every integer $k\ge1$ and every $\varepsilon>0$, there exists $r_0=r_0(k,\varepsilon)$ such that every $r$-regular graph on $n$ vertices with $r\ge r_0$ contains a $k$-regular subgraph covering at least $(1-\varepsilon)n$ vertices. Previously, the conjecture was known for $k\in\{1,2\}$ and for $k$ and $r$ both even. We answer this question affirmatively for general $k$ and $r$ with $\varepsilon=O_k(r^{-1/2})$. For $k=2$ and every odd $r\ge3$, we show that $\varepsilon=1/(r^2-3)$ suffices which is best possible.

Coronoid systems, nanotubes and toroidal polyhexes (or fullerenes) can all be regarded as carbon networks composed of carbon atoms linked in hexagonal shapes. The resonance graphs of coronoid systems and nanotubes are not necessarily connected. For coronoid systems and elementary nanotubes, by using flow across cuts the present authors gave criteria for two perfect matchings lying in the same connected component of the resonance graph (Discrete Appl. Math. 395 (2026) 443-455). However, the sufficiency of such criterion does not hold for general nanotubes and toroidal polyhexes. In this paper we strengthen this requirement to obtain valid criteria for two perfect matchings of a nanotube (resp. toroidal polyhex) to lie in the same connected component of its resonance graph: they have the same flows across cuts along the $x$-axis (resp. longitude and latitude) and the same ladders. For toroidal polyhexes, our method uses homotopic classes of simple loops on the torus, and the above criterion can be simplified by using only simple flows, for the case in which two perfect matchings have alternating hexagons.