Papers

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz–Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz–Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

For fixed nonnegative integers $m_1,m_2$, let $A_n=(a_{j-i})_{i,j=1}^n$ be the leading $n\times n$ section of a Toeplitz matrix with lower and upper semibandwidths $m_1$ and $m_2$. We give two constructive Laplace-expansion methods for scalar recurrences of $\det A_n$ and $\perm A_n$. The increasing-rows method eliminates a fixed family of boundary cofactors and gives recurrence order at most $d=\binom{m_1+m_2}{m_1}$ for both sequences. The row-column method closes normalized boundary minors recursively and packages them in a sparse transfer matrix. Its reachable states are classified exactly: level $j$ is indexed by a pair of $j$-subsets of $[m_1]$ and $[m_2]$. Hence the transfer dimension is $d$, and we obtain an explicit formula for the number of nonzero transitions. For determinants, the complementary cofactors of the increasing-rows construction are coordinates of the classical compound companion representation. The independently constructed row-column transfer has the Widom characteristic polynomial and is generically similar to the compound transfer. Thus the order $d$ recurrence is generically minimal for the unrestricted fixed-band determinant family. For permanents the same state graph gives the binomial upper bound, without a general minimality claim. The pentadiagonal case recovers Sweet's order-six determinant recurrence and its permanent analogue, while the one-superdiagonal family gives closed scalar recurrences and rational generating functions. Position-dependent band weights preserve the finite state graph but replace the constant transfer by a cocycle. For cyclic determinants, Fourier diagonalization produces all subset products of the symbol roots and a generically minimal annihilator of degree $2^{m_1+m_2}$, corresponding to the passage from one exterior degree to the full exterior algebra.

For a fixed integer $k\ge 2$, let $W_{2k+1}=K_1\vee C_{2k}$ be an odd wheel graph. The fixed-size spectral extremal problem aims to determine \[ \operatorname{spex}(m,W_{2k+1}):=\max\{\rho(G): e(G)=m,\ G \text{ is } W_{2k+1}\text{-free}\}, \] where $\rho(G)$ denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [18] proposed the following conjecture: For large $m$, every $W_{2k+1}$-free graph of size $m$ satisfies $\rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2}$ with equality precisely for $K_k\vee qK_1$ and $m-\binom{k}{2}=kq$. Very recently, Fang, Zhai and Zhang [7] confirmed the Yu–Li–Peng conjecture. When $m$ is large, $k\ge 2$ and $m-\binom{k}{2}$ is not divisible by $k$, the exact solution for the above problem is still open. Regarding this problem, Yu, Zhang, and Zhang [19] proposed the following conjecture: Let $r$ be a nonzero remainder when $m-\binom{k}{2}$ is divided by $k$. Then $S_{k,m}$ is the unique graph among $W_{2k+1}$-free graphs of size $m$ having maximum spectral radius, where $S_{k,m}$ is obtained from $K_k\vee qK_1$ by adding a vertex $z$ and joining it to exactly $r$ vertices of the $K_k$. In this paper we address this problem in each nonzero residue class. Our result completely settles the Yu-Zhang-Zhang conjecture for $k\ge 3$.

We classify the polyominoes obtained by adjoining four straight arms to a single square, allowing zero arm lengths, according to their ability to tile rectangles, half-strips, bent strips, quadrants, strips, half-planes, and the plane. We also classify their ability to tile an integer enlargement of themselves. Tiles occupy whole square-grid cells; translations, rotations, and reflections are permitted. Exactly five capability profiles occur. For the family $P(n,1,1,0)$, the rectangle profile holds for $n\le3$ and the bent-strip profile, with no half-strip or rep-tiling, for every $n\ge4$. A cross with four positive arms tiles the plane precisely when two opposite arms have length one; it never tiles a half-plane. Explicit periodic constructions and geometric obstructions are combined with finite symbolic case certificates. A Lean 4 development verifies the full classification for every natural four-tuple, including the interpretation of the certificates as statements about arbitrary infinite tilings. The account incorporates the author's 2020–2021 L- and T-polyomino work, reconstructs Dahlke's gun argument, and documents the subsequent AI-assisted proof development and formalization.

Gross, Mansour, and Tucker~[European Journal of Combinatorics, 95 (2021): 103329] introduced the partial-twuality polynomials of ribbon graphs. Recently, Deng, Jin, and Yan generalized the partial-twuality polynomials to the framework of matrix algebra and investigated several of their basic properties. They asked whether there exist matrix operations, called partial duality $\delta$ and partial Petrie duality $\tau$, on pairs $(M,A)$, where $M$ is a square matrix whose rows and columns are indexed by a finite set $V$ and $A\subseteq V$, such that $\delta^2=\tau^2=(\delta\tau)^3=id$ and the exponent of the partial-twuality polynomials coincides with some parameter of the matrix obtained by applying \(\bullet\) to \((M, A)\). In this paper, we introduce a paired-matrix framework for partial-twuality polynomials over the binary field $\mathbb{GF}(2)$. We prove that there exist two local operations \(\delta\) and \(\tau\) on \(\bigl((M,I_{|V|}),A\bigr)\) satisfying $\delta^2=\tau^2=(\delta\tau)^3=id$ and $P_{\langle \bullet \rangle}((M,I_{|V|}),z)=P_{\langle \bullet \rangle}(M,z)$ for $\bullet \in \{\delta, \tau, \delta\tau, \tau\delta, \delta\tau\delta\}$, thereby answering their question affirmatively. Finally, we establish a recurrence relation for the partial \(\langle\delta\tau\delta\rangle\)-polynomial with respect to an edge. This recurrence enables the computation of the partial \(\langle\delta\tau\delta\rangle\)-polynomial for certain bouquets, simple graphs, and simple signed graphs.

For all prime powers $q\geq5$, we construct lattices $\mathcal{L}_q\subseteq\mathbb{Z}^q$ with kissing numbers \[ \tau(\mathcal{L}_q)\geq \left(\frac{1}{2\pi e^2}+o(1)\right)\sqrt{q}\,e^{2\sqrt{q}}. \] The same asymptotic bound holds on a set of integer dimensions of natural density $1$, and in every sufficiently large integer dimension $n$ with an additional factor $e^{-\tfrac{1}{2}n^{1/40}}$. The construction is an extension of a previous construction by Bennett-Peikert based on Reed-Solomon codes.

Let $\Gamma^s(2,q)$ be the graph induced in the Grassmann graph by the $q$-ary simplex codes of dimension 2. For $q=4$, this graph is known to have diameter 3. We prove that the same diameter occurs for every prime power $q\geq5$. Together with the elementary cases $q=2,3$, this gives diam$\Gamma^s(2,q)=0,2,3$ for $q=2$, $q=3$, and $q\geq4$, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For $q\geq5$ an explicit diagonal pair of simplex lines gives the matching lower bound; we give two proofs, one using moments and one using products. For $q\geq7 we also retain an independent counting proof. The case $q=4$ is handled separately.

We study the mixing behavior of three q-deformed card shuffles on $\mathcal{H}_q(S_n)$: the short systematic scan introduced in \cite{DiaconisRam2000}, the $q$–deformed random–to–random shuffle introduced in \cite{AxelrodFreedBraunerChiangComminsLang2024}, and the $q$–deformed $k$–star transposition shuffle, whose $q=1$ counterpart was studied in \cite{ArfaeeNestoridi}. The first two chains were diagonalized in \cite{DiaconisRam2000} and \cite{AxelrodFreedBraunerChiangComminsLang2024}, respectively. Here, we diagonalize the $q$–deformed $k$–star transposition shuffle and use the spectra of the three chains to study their mixing behavior. For fixed $q>1$, we prove that all three exhibit total variation and $\ell^2$ cutoff. For the short systematic scan, these improve both the upper and lower bounds of Diaconis and Ram \cite{DiaconisRam2000}.

Stable polynomials are the natural multivariate generalization of real rooted univariate polynomials. While their definition is purely algebraic in nature, they have deep connections to combinatorics. One such connection is the support theorem proved by Brändén in 2007, showing that the support of any stable polynomial is a jump system, and hence that the support of any multiaffine stable polynomial is a $\Delta$-matroid. In this work, we generalize this result, showing that coefficients of multiaffine real stable polynomials give rise to oriented $\Delta$-matroids, and coefficients of multiaffine stable polynomials over Puiseux series give rise to valuated $\Delta$-matroids.

At a 2021 AIM workshop, Guo conjectured that the positive square energy s+ = E+_2 should inherit the familiar edge-addition monotonicity of the spectral radius, rho(G + uv) >= rho(G). That conjecture was subsequently shown to fail at p = 2. Tang, Liu, and Wang then introduced positive p-energy, proved nonmonotonicity for every 1 <= p < 3, and in version 3 of their preprint (26 March 2025) explicitly conjectured that monotonicity should hold for p >= 3. We disprove this conjectured high-exponent extension completely: for every real p > 2 there are infinitely many connected graphs G and nonedges uv such that E+_p(G + uv) < E+_p(G). Together with the Tang-Liu-Wang counterexamples below 3, this shows that no exponent p >= 1 restores the spectral-radius-style monotonicity: positive p-energy can decrease under the addition of an edge for every real p >= 1. The construction is a chain of clique blocks joined by regular bipartite graphs. An equitable quotient converges to Q = I + cA(P_k). For noninteger p, a binomial-series sign argument for a fractional power of I - cA(P_k) gives the required negative endpoint entry. At integer exponents, choosing c across the first spectral threshold leaves exactly one negative eigenvalue, and path locality forces the positive spectral contribution to have negative sign. We also give a fully rational 38-vertex certificate at p = 4 and determine the complete failure interval of a fixed 17-vertex counterexample at p = 3.

Let $T$ be a tree. Stanley asked whether the chromatic symmetric function $X_T$ determines $T$ up to isomorphism. We approach this open problem by regarding $X_T$ as a polynomial in the power-sum symmetric functions $p_1, p_2, \dots$ and studying the invariant $\Phi_T = (\partial X_T/\partial p_1)|_{p_1 = 0}$. We prove that $\Phi_T$ distinguishes every proper tree whose weighted skeleton, the tree obtained from $T$ by weighted contraction of all leaf edges, has distinct weights at non-leaf vertices. We prove further an equivalent formulation of Stanley's question obtained by attaching a fixed positive number of leaves to every vertex of a tree. Finally, we count spanning forests with at most $t$ edges, grouping them by the sizes of their connected components. We prove that these counts cannot distinguish all trees on $k \ge 4$ vertices unless $t \ge \lfloor k/2 \rfloor$.

We develop a geometric framework for distance-based formation control that separates the evolution of inter-agent distances from its realization by compatible node motions, reducing the stability problem to the edge space. We show that local exponential convergence of the edge dynamics implies local exponential convergence of the formation, and that stability is certified by spectral properties of a linear edge operator. We introduce a hierarchy of generic spectral properties — weak admissibility, admissibility, and strong admissibility — that provide necessary conditions for local exponential stability. Specializing to directed sensing, we obtain a necessary and sufficient spectral condition for local stability at an arbitrary target, together with a quadratic sufficient certificate. These conditions reveal that stability depends jointly on the graph orientation and target geometry, and show that persistence is neither necessary nor sufficient for local convergence. We show that every generically rigid graph admits an admissible orientation and, for acyclic orientations, we give an exact combinatorial characterization of admissibility. Finally, the quadratic certificate leads to a semidefinite program for synthesizing stabilizing edge gains.

In a 2021 paper, Jankiewicz, Norin, and Wise asked whether there exists a finite $4$-connected graph of girth at least four and nonnegative Charney–Davis curvature such that no $4$-connected ordinary subgraph admits a legal system. We construct such a graph by starting from the hexagonal prism and attaching three $K_{3,4}$-based caps along pairwise disjoint induced $4$-cycles. The key structural input is a restriction theorem showing that a legal system on an induced-$4$-cycle amalgam restricts to each side, so the obstruction carried by the negatively curved prism survives the attachments. The resulting $33$-vertex graph is $4$-regular and $4$-connected, has girth four and Charney–Davis curvature one, and, by $4$-regularity, is its own unique $4$-connected ordinary subgraph.

Gao and Xie conjectured that the coefficients of the ordinary, unnormalized inverse Kazhdan-Lusztig polynomial of every matroid are log-concave with no internal zeros. We give counterexamples and prove locality and composition formulas for deletions of projective charts. For every finite field and every nonempty finite set S contained in {2, 3, ...}, these formulas yield simple 3-connected matroids whose negative internal Turán determinants occur exactly at S. Their coefficients are positive and strictly decreasing. The polynomial degree can be prescribed above max S, and all selected Turán ratios can tend to zero simultaneously. The universal first-index inequality remains open.

Pate proved ordinary irreducible-immanant permanental dominance through order $13$ and identified $(4,4,3,3)$ as the sole remaining order-$14$ case, with $(5,4,3,3)$ and $(3^5)$ forming the order-$15$ frontier. These three cases are settled here; consequently $d_\lambda(A)/f^\lambda\le \operatorname{per}(A)$ for every partition $\lambda\vdash n$ with $n\le15$ and every complex Hermitian positive-semidefinite matrix $A$. The argument also yields results beyond this finite frontier: an exact four-term bridge for $(4,4,3,3)$, the uniform family $(m,4,3,3)$, a two-parameter family $(a,b,3,3)$ for $a\ge b\ge4$ and $5a\ge8b$, and a long-first-row criterion for arbitrary fixed tails. These results arise from explicit specializations of Pate's $W$-function positivity framework using partial swaps, Young projectors, Pieri–content identities, and branching data. For $(3^5)$, an exact Farkas certificate shows that the central-projector partial-swap cone is insufficient; a branching-refined one-swap construction escapes this obstruction and yields a positive $106+19$-witness certificate. Boundary-compression and node-moving results further describe the reach and limitations of the local-filter method. All finite certificates are checked by exact integer or rational arithmetic and are supplied as ancillary material. The order-$14$ bridge is additionally formalized and kernel-checked in Lean 4 for all complex Hermitian positive-semidefinite matrices, including the exact coefficient normalization and the deduction of $(4,4,3,3)$ permanental dominance from four explicitly stated Pate inequalities.

A cycle system for a matroid $M$ is a collection of cycles (unions of circuits) whose intersection properties mimic the cut sets of a graph. Cycle systems were introduced by Corry, the first author, McClain, Perkinson, and Yi, who showed that the $h$-vector of any matroid that admits a cycle system is a pure $O$-sequence, confirming a conjecture of Stanley for this class. Those authors also conjectured that any matroid admitting a cycle system must be binary. Here we answer this conjecture in the affirmative, and prove the stronger result that any such matroid must in fact be regular (representable over any field). From this, we conclude that if $M$ is connected, every cycle system for $M$ is a basis for its circuit space of. We establish other properties of cycle systems along the way, which may be of independent interest.

For every finite set $A\subset\mathbb{Z}$, we construct a bijection $p:\mathbb{N}_0\to\mathbb{Z}$ beginning with $0,1$ that contains no four-term arithmetic progression, in occurrence order, whose first value belongs to $A$. More generally, for each $n\ge2$ the enumeration can begin with $0,1,3,\ldots,2^{n-1}-1$ and simultaneously avoid every such progression starting in $A$ or among these first $n$ entries. The construction uses finitely branching prerequisite relations and an explicit bounded integer potential. This proves finite prerequisite closure and gives an exhaustive enumeration, rather than merely a total order. We also give an extension theorem for finite prefixes with compatible binary-tail constraints. These results do not determine whether every enumeration of the integers contains an ordered four-term arithmetic progression.

Clustering systems provide a natural way to encode structural information contained in phylogenetic networks. In this note, we study the clustering systems of normal and tree-child networks through an overlap-based property of set systems, called not-overlap-covered (NOC). We show that the NOC property is equivalent to inclusion-visibility, a memberwise formulation of the strict-compatibility condition previously used for tree-child clustering systems. We characterize normal networks as precisely the semi-regular networks whose clustering systems satisfy NOC. Consequently, a clustering system is realized by a normal network if and only if it satisfies NOC, or equivalently, if every one of its clusters is inclusion-visible. In this case, the Hasse diagram provides a canonical normal realization. These are exactly the clustering systems realized by tree-child networks. The NOC formulation yields a sharp quadratic upper bound on the number of distinct clusters of tree-child and normal networks and a direct polynomial-time recognition algorithm. Finally, we explore several consequences of the NOC perspective beyond the phylogenetic setting. These include connections to the enumeration of normal networks, an order-theoretic interpretation of inclusion-visibility, structural properties of NOC set systems, and a tractable special case of Minimum Set Cover, which is NP-hard in general.

Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cycles $K_n$. Write $\lambda(s)=\log\{\Gamma(1+s)/\Gamma(1+s/2)^2\}$, let $s_\kappa$ minimize $(1+\kappa\lambda(s))/s$ on $(0,\infty)$, and put $v(\kappa)=\kappa\lambda'(s_\kappa)$ and $a_\theta=\lambda(s_\theta)/s_\theta$. Under the Ewens measure with any fixed parameter $\theta>0$ we prove $M_n=v(\theta)\log n+a_\theta(K_n-\theta\log n)+O_P(\log\log n)$, so that the standardized pair $(K_n,M_n)$ converges jointly to $(G,G)$ with $G$ standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on $\theta$: for every compact $[\kappa_-,\kappa_+]\subset(0,\infty)$ there is a finite $C$ such that $P(|M_n-v(k/\log n)\log n|>C\log\log n \mid K_n=k)$ tends to $0$ uniformly over integers $k$ with $\kappa_-\log n\le k\le\kappa_+\log n$, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than $n/(\log n)^4$ are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.

Given a graph $F$, a family $\mathcal F$ of graphs on $[n]$ is \emph{$F$-intersecting} if $G\cap H$ contains a copy of $F$ for every $G,H\in\mathcal F$. We prove that there exists an absolute constant $\varepsilon>0$ such that every $P_4$-intersecting family $\mathcal F$ satisfies $|\mathcal F|\le\left(\frac12-\varepsilon\right)2^{\binom n2}$, which resolves a conjecture of Alon. Combined with Alon's reduction, this proves that a graph $F$ admits $F$-intersecting families of asymptotic density $1/2$ if and only if $F$ is a star forest. Multi-colored versions are also considered.

In 2021, Chern and Tang introduced two families of two-parameter modular functions associated with the Rogers-Ramanujan continued fraction. They established recurrence relations that express the members of these families in terms of eta quotients, and used these expressions to obtain dissection formulas. Motivated by their work, we construct a three-parameter extension and derive the corresponding recursive eta quotient representations. Our result also has applications to dissection formulas. In particular, it is used in separate work to obtain $5$-dissection formulas for overpartitions with restricted odd differences.

Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $\Delta\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $\Delta$. We prove that \[ |A|\ll_{\Delta,d} N^d\exp\!\left(-c_{\Delta,d}\sqrt{\log N}\right) \] improving upon a polylogarithmic bound due to Magyar. We perform a density increment argument using the circle method, and we introduce a “cut operator” method to decouple the weighted exponential sum over the system of quadratic forms describing the simplex. Our proof combines ideas from graph theory, functional analysis, and the geometry of numbers. In the process, we apply Finner's fractional form of Hölder's inequality, the analytic large sieve, and Kim's mean value formula for primitive lattice flags.

Let $g(n)$ denote the minimum number of peeling sequences among all $n$-point sets in general position in the plane. Dumitrescu and Tóth proved an exponential upper bound with base $12.29$, and Simon subsequently lowered the base to $9.78$. Using the same recursive construction, we prove \begin{equation*} g(n) \le (8+o(1))^n. \end{equation*}

We show that every finite premaniplex of rank $3$ or $4$ is the symmetry type graph of a finite maniplex, settling the finite rank $3$ and $4$ case of the maniplex version of the symmetry type graph problem. The proof uses the fact that the universal string Coxeter groups of rank $3$ and $4$ are amalgams (of finite groups), hence they act on a tree, which allows us to use a lifting theorem of Potočnik and Spiga.

We discuss dominant representations (or Young diagrams), in thermal matrix models with gauge symmetry from the perspective of partial deconfinement. We propose a prescription for defining the dominant representations for thermal matrix models with interaction terms. As an explicit example, we consider the large-$N$ Gaussian matrix model. We obtain the Vershik–Kerov–Logan–Shepp (VKLS) shape of the dominant Young diagrams through a new analytic saddle-point analysis based on a mapping of the representation theory of U($\infty$) to free fermions in two spacetime dimensions. This computation provides a direct derivation of the previously observed functional relation between the shape of the dominant Young diagrams and the eigenvalue distribution of the thermal holonomy: the position of the complex saddle point is naturally identified with the eigenvalue. The dominant Young diagrams admit a natural interpretation in terms of partial deconfinement: the number of rows in the dominant Young diagrams matches the size of the submatrix corresponding to the deconfined subsector.

We develop a projection and fibering method for sets of small combinatorial doubling in (not necessarily abelian) discrete groups. As an application, in the abelian case we prove that, if $A$ is finite, the ambient group has exponent $r$, and $|A+A|\leq K|A|$, then \[ |\langle A\rangle|\leq r^{(2+o(1))K}|A|. \] This answers a question of Ruzsa with an optimal leading coefficient, independent of Fox–Pham. The main ingredient is a discrete version of a fiber spillover argument. For sets in $2$-step nilpotent groups of exponent $r$, we also prove that $|A^3|\leq K|A|$ implies $|\langle A\rangle|\leq r^{(2+o_K(1))K}|A|$. The proof combines the abelian theorem with a weighted averaging of central fibers and commutators.

An old result of Tutte states that any $d$-regular graph contains a spanning subgraph in which every vertex has degree $k$ or $k+1$, for every $1\leq k\leq d$. We generalize this statement to hypergraphs, showing, for example, that every $3$-uniform $d$-regular hypergraph contains a subgraph in which all degrees are $k, k+1$ or $k+2$, for every $1\leq k\leq d$. This statement is best possible in the sense that the corresponding statement with only two allowed consecutive values is not true. We provide generalizations of this statement to higher uniformities and discuss several open problems.

We show that the rowmotion operator from dynamical algebraic combinatorics behaves the same on all alt $\nu$-Tamari lattices for a fixed lattice path $\nu$. We use this invariance of rowmotion to establish cyclic sieving and homomesy results for rational Tamari lattices. We also conjecture that this invariance extends to the more general cross Tamari lattices.

We construct a permutation of the positive integers containing no four-term arithmetic progression as a subsequence.

For every positive integer $k$, we prove that the homogeneous cut semigroup of every $K_5$-minor-free graph is saturated at height $k$ if and only if every planar $k$-graph is $k$-edge-colourable. Here a $k$-graph is a loopless $k$-regular multigraph in which every odd vertex cut has size at least $k$. A triangle expansion of a cubic plane dual converts cut decompositions into perfect-matching decompositions. The converse uses symmetric difference with a fixed perfect matching. The equivalence identifies the normality conjecture for $K_5$-minor-free cut polytopes with Seymour's planar edge-colouring conjecture. In particular, the known cases $k\leq 8$ give saturation through height eight.