Papers

Given a subvariety of the Grassmannian and a data point, we seek to find a point on the subvariety minimizing the Euclidean distance to the data point. The number of complex critical points of this optimization problem is the Euclidean distance (ED) degree. We show that the ED discriminant of the whole Grassmannian, that is, the set of data points with a number of critical points different than the ED degree, is the discriminant of the characteristic polynomial of the data as a projection matrix. Another closely connected algebraic complexity measure for a subvariety in the Grassmannian is the Grassmann distance (GD) degree, which is the number of complex critical points of the distance optimization problem when the data point itself is in the Grassmannian. We give formulae for ED and GD degrees of geometrically meaningful subvarieties of the Grassmannian, namely, products of smaller Grassmannians, matroid realization varieties, and Schubert varieties.

Building on our classification of shape-Wilf-equivalence classes for triples of patterns of length three, we complete the classification for quadruples and quintuples. The larger pattern sets exhibit structural features that are not captured by the encoding methods used for triples and require additional combinatorial tools. Our main new ingredient is a Dyck-path approach to Ferrers boards containing the staircase board. By introducing distinguished cells and related structural parameters on the associated Dyck paths, we obtain new shape-Wilf-equivalences and, for several classes, explicit formulas for the number of avoiding transversals on a fixed Ferrers board. We further study the corresponding pattern-avoiding perfect matchings through the pattern-preserving bijection of Bloom and Elizalde. Besides deriving recurrence relations, we give bijective interpretations of several matching enumeration sequences in terms of grand Dyck paths, Schröder paths, and Dyck paths. These constructions provide additional combinatorial structure beyond the classification itself. Together with the companion paper on triples, the present work completes the classification of shape-Wilf-equivalence classes for subsets of patterns of length three, apart from the trivial empty and full pattern sets.

The linear list Hadwiger conjecture asserts that every $K_t$-minor-free graph is $O(t)$-choosable. Following an approach outlined by Delcourt and Postle, we recently established $O(t\log\log t)$-choosability. Soon afterward, OpenAI provided a proof of the conjecture. A key new ingredient of their proof is a linear bound on the choice number of $K_t$-minor-free graphs with at most $t^{11/10}$ vertices. In this paper, we give a relatively simple proof of this small-graph bound for graphs with at most $t(\ln t)^5$ vertices.Combining this with the argument from our earlier $O(t\log\log t)$ bound, we obtain a shorter and somewhat more accessible proof of the conjecture.

We give a necessary and sufficient condition for a combinatorial D-class of a finite semigroup to have a permutation matching: that its structure matrix admit a balanced weighting, that is, non-negative weights with constant row sums and constant column sums. This weakens the regularity hypothesis, the role of Hall's lemma being taken up by the Birkhoff–von Neumann theorem. We then apply it to show that the monoid of all order-preserving self-maps of an n-element chain, has a permutation matching.

In distributed storage systems, redundancy enables a single data object to be reconstructed using several disjoint groups of servers. The resulting service rate region (SRR) captures all combinations of object request rates that the system can support simultaneously without overloading any individual server. We analyze the SRR of binary codes obtained via the Plotkin-type construction. We demonstrate that each recovery set for an information object of the Plotkin-type code $\mathcal{C}_p$ induces a corresponding recovery set for the same data object characterized by the underlying code $\mathcal{C}$. Furthermore, by characterizing the recovery sets structure of $\mathcal{C}_p$ in terms of the recovery set structure of $\mathcal{C}$, we identify the additional recovery sets introduced by this construction. Using the associated recovery hypergraphs, we establish bounds on the SRRs of $\mathcal{C}_p$ and iterated code $\mathcal{C}_{p^m}$ in terms of the SRR of $\mathcal{C}$. We consider the family of first-order binary Reed-Muller codes and show how the recovery structure, and consequently, the service rates of R$(1,m)$ for arbitrary $m$ can be recursively derived from a generator matrix of R$(1,2)$ through our results for successive Plotkin-type constructions.

Motivated by Kawanoue's construction with the finite-sum formulas, we construct an explicit homogeneous basis for the derivation module of the cone of the extended Catalan arrangement of type Dl. We further show that the restriction of this basis at infinity coincides, up to a nonzero constant, with the integral basis constructed by Feigin-Wang-Yoshinaga. This identifies the geometric motivation behind the parameter choices.

Liftability was first introduced for paving matroids, where it was used to systematically construct polynomials in the ideals of the associated matroid varieties. In this paper, we generalise the notion of liftability to matroids with realisable truncations. We additionally introduce the notion of generic realisability of a matroid; a strong notion of realisability that implies both realisability and liftability of the matroid. We study the behaviour of liftability and generic realisability under the weak order of matroids within the lattice of matroids with fixed truncation. We develop a unified framework for studying liftability of matroids through Oxley–Wang derived matroid of generic matroid realisations. We characterise liftability and generic realisability in terms of Oxley–Wang derived matroids. Moreover, we give an algebraic description of the bases of the generic Oxley–Wang derived matroids in terms of coordinate sections of matroid realisation spaces and relate our constructions to slack realisation spaces of matroids. As an application of our framework, we obtain a combinatorial characterisation of liftable and generically realisable paving matroids and connect it with Whiteley's theory of scene analysis. In particular, our approach gives a new proof, via Oxley–Wang derived matroids, of Whiteley's characterisation of hypergraphs admitting non-trivial (sharp) scenes. Finally, we compute the generic Oxley–Wang derived matroid for a special class of sparse paving matroids.

A dominating set $D$ of a graph is called $1$-limited if every vertex of $D$ has at most one neighbor outside $D$, while a $(1,2)$-dominating set is a dominating set in which every vertex of the set has at least two neighbors within the set. These two notions coincide on cubic graphs. We prove that the decision problem 1-Limited Dominating Set is $\mathsf{NP}$-complete even when restricted to $2$-connected planar cubic graphs, thereby completing the known complexity results for $k$-Limited Dominating Set for all fixed positive integers $k$. We also determine the exact $1$-limited domination number of the entire Goldberg family. This provides a further infinite family of cubic graphs supporting several open conjectures and proposed bounds concerning $(1,2)$-domination and induced cycles.

Let $p$ be an odd prime, let $m\geq1$ be an integer, and let $Q$ be a nondegenerate quadratic form on $\mathbb F_p^{2m}$. For a nonempty set $E\subseteq\mathbb F_p^{2m}$, write $\Delta_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $|E|>p^m$, \[ |\Delta_Q(E)|\gg \min\left\{\frac{p}{L_E},\, \frac{|E|}{p^{m-1}L_E^2}\right\}, \qquad L_E=\log\left(2+\frac{p^{m+1}}{|E|}\right), \] with an absolute implied constant independent of $p$, $m$ and $Q$. This confirms the Erdős–Falconer distance conjecture up to a logarithmic factor. In the anisotropic planar case $m=1$, for small sets, we also prove that \[ |\Delta_Q(E)|\gg\frac{|E|}{\log(2|E|)}, \qquad (1\leq |E|\leq p), \] which is also optimal up to a logarithmic factor.

We prove that if the product of two class transpositions has finite order, then its order divides 840. This gives a complete affirmative answer to Question 18.48 in the Kourovka Notebook. We also obtain a criterion for the product of two class transpositions to have infinite order. In addition, we construct a pair of class transpositions whose product has order 840, showing that the bound is sharp.

We study three recurrent double sequences whose colored diagrams encode mirrored versions of the finite-ordinal construction. Two sequences use three letters and realize variants with two and three initial symbols. A four-letter sequence realizes the von Neumann construction from one initial symbol. For each sequence we describe its cells explicitly and identify the ordered nesting of its rectangles. For four letters, we distinguish ordinary and virtual boxes and prove their assembly rules. The proofs proceed by induction along antidiagonals. For the constructions with one and three initial symbols, we reduce the recurrence to finite sets of local configurations and verify those sets by computer. We derive formulas for the numbers of nested boxes, including arithmetic-term representations, and we prove that none of the three sequences is $k$-automatic for any integer $k \geq 2$.

Two integer partitions $t$-intersect if they have at least $t$ common parts, counted with multiplicity. We study the largest $t$-intersecting families of integer partitions of $n$ into exactly $k$ positive parts. The canonical $t$-star consists of the partitions containing at least $t$ ones. Applying Kupavskii's weak-spread theorem, we prove that this star is largest whenever $n\ge Ak^3$, for every fixed $A>24$ and all sufficiently large $k$, uniformly over $1\le t<k$. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large $k$ and for every $1\le d<k$, one may choose $d/4\le t\le d$ and $n=\lfloor tk^2/3\rfloor$ so that a $t$-intersecting family is strictly larger than the canonical star.

We show, under a certain assumption, that the Cheeger constant of a connected Cayley graphon is bounded from above in terms of its bipartiteness ratio, and also in terms of the gap between the top of the spectrum of its Laplacian and $2$. Using this result, we prove that if a connected Cayley graphon $C(G,\gamma)$ is bipartite, then there exists an index two subgroup $H$ of $G$ that avoids the support of $\gamma$.

A storage code on a graph assigns a symbol to each vertex so that the symbol can be recovered from its neighbors. We prove that the binary full-parity storage codes on triangle-free coset graphs of primitive BCH codes have rate tending to one for every fixed error-correction capability of at least two. The conclusion also holds for an increasing error budget under an explicit growth condition. The proof bounds the binary rank of convolution matrices associated with polynomial maps. Low coordinate degree forces cancellation in the polynomial representing the image, and a polynomial rank bound converts this cancellation into a quantitative estimate for the storage rate. For the double-error-correcting family over $\mathbb{F}_{2^m}$, we prove that the storage deficiency is $\Theta(((1+\sqrt{5})/4)^m)$. This refinement follows from an exact Fibonacci formula for a polynomial coefficient rank and a matching exponential lower bound after finite-field evaluation.

The $k$-Schur functions (resp., $K$-$k$-Schur functions) are the symmetric function representatives of Schubert classes in the homology (resp., $K$-homology) of the affine Grassmannian of type $A$. We prove that every $K$-$k$-Schur function is a nonnegative integer linear combination in the basis of $k$-Schur functions. This settles a conjecture by Lam, Schilling and Shimozono. The proof relies on realizations of $k$-Schur and $K$-$k$-Schur functions as Catalan and Katalan functions, respectively.

We classify the set of finite simple undirected graphs whose corresponding graph operator system is C*-nuclear. We prove that the operator system associated to the graph is C*-nuclear if and only if the graph is 2-chordal, as defined by McCullough. In particular, we present the first known example of a non-chordal graph whose graph operator system is C*-nuclear, as well as the first known examples of graph operator systems which are not C*-nuclear.

Jin, Milojević, Tomon and Zhang established a powerful recursive estimate relating the positive eigenvalues of a graph whose least eigenvalue is small in absolute value. We establish a refinement of this result, yielding improved estimates across spectral graph theory and discrepancy theory, and for Chowla's cosine problem. Recent results of Janzer, Tomon and Yip allow us to directly transfer least eigenvalue estimates to the corresponding surplus estimates. Using these tools, we fully resolve a conjecture of Räty, Sudakov and Tomon. We show that, for an $n$-vertex graph $G$ with least eigenvalue $\lambda_n$ and surplus $\operatorname{sp}(G)$, if $G$ is $\epsilon$-far from all disjoint unions of cliques, then $|\lambda_n|\geq \Omega_\epsilon(n^{1/4})$ and $\operatorname{sp}(G)\geq \Omega_\epsilon(n^{5/4})$. Furthermore, we show that when $G$ is $n^{-o(1)}$-far from all disjoint unions of cliques, we have $|\lambda_n|\geq n^{1/4 - o(1)}$ and $\operatorname{sp}(G)\geq n^{5/4 - o(1)}$. Finally, we show that the surplus of a $K_t$-free graph with $m$ edges is at least $m^{0.614 - o(1)}$ as $m$ tends to infinity.

We prove that the product of chains $\mathbf m\times\mathbf n$ is not Schur positive whenever $n\ge4$ and $m\ge3n-1$. Writing $m=n+k$, we exhibit a negative Schur coefficient indexed by $(2n+k-1,2n+k-3,\ldots,k+5,k,k,4)$ and evaluate it explicitly as $(n-2)!$ times a polynomial of degree three in $k$. Together with earlier results, this establishes the two-case interpretation of the conjecture of Li, Qiu, Yang, and Zhang: these products are not Schur positive when $n=2$ and $m\ge8$, or when $n\ge3$ and $m\ge n+5$.

Structural system theory studies which network topologies can sustain a prescribed system property such as controllability or stability. When the topology is itself random, the relevant question becomes probabilistic: how likely is a graph drawn from a stochastic model to sustain the property? Such probabilities measure the abundance and robustness of the property across topologies, and indicate whether systems requiring it can be reliably deployed in uncertain environments. We address this question for asymptotic stability of linear systems in the directed graphon setting. We consider two graph-theoretic properties. The first is $\mathcal N$, and it requires that for every $k\leq n$ some $k$-vertex induced subdigraph of $D$ admits a cycle cover, and the second is $\mathcal S$, which requires that these subdigraphs can be chosen so that their node sets form a nested sequence $V_1\subset\cdots\subset V_n=V(D)$ starting from a single vertex with a loop. We have shown that $\mathcal N$ is necessary and $\mathcal S$ is sufficient for structural stability. We sample $D$ from a directed step-graphon $W$. Our main results give necessary and sufficient conditions for $\Pr(\mathcal N)\to 1$ and $\Pr(\mathcal S)\to 1$ as $n\to\infty$. In more detail, to a step-graphon $W$ with skeleton digraph $S$ on $q$ nodes and concentration vector $x^*$ we associate a cycle polytope $\vec{\mathcal X}(S)\subseteq\Delta_q$. The conditions are then formulated in terms of the position of $x^*$ within $\vec{\mathcal X}(S)$, the dimension of the polytope, the loop density of $W$ and, for $\mathcal S$, an ordering condition on the cycles of the skeleton. Together these results identify, for directed step-graphons, the regime in which a sampled topology is overwhelmingly likely or unlikely to sustain stable dynamics.

We study the average-case matrix discrepancy problem: given independent normalized $d\times d$ Gaussian orthogonal ensemble matrices $A_1,\dots,A_N$ and a fixed margin $\kappa>0$, find signs $\sigma_1,\dots,\sigma_N\in\{-1,1\}$ such that the operator norm of $\sum_{i=1}^N \sigma_i A_i$ is at most $\kappa\sqrt{N}$. Focusing on the proportional regime $N/d^2\to \tau\in(0,\infty)$ as $d\to\infty$ followed by the small-margin limit $\kappa\downarrow 0$, we characterize the density required by stable offline algorithms and by online algorithms. In the offline setting, we construct a polynomial-time recenter-and-round algorithm that is noise-stable and succeeds whenever $\tau=\Omega(\frac{1}{\kappa^2\log(1/\kappa)})$, along with a matching lower bound for all stable algorithms. In the online setting where each sign must be chosen irrevocably upon observing the corresponding matrix, we determine the exact limiting performance of the Frobenius-greedy algorithm, establishing that it succeeds when $\tau>\tau_{\rm FG}(\kappa)\sim \frac{\pi}{4\kappa^2}$, as well as a matching lower bound for all online algorithms by conditioning on a revealed prefix. At the core of our algorithms lies rotational symmetry, which enables us to transfer Frobenius norm control into operator norm guarantees. Together, our results identify the algorithmic phase transition points for random matrix discrepancy: $\Theta(\frac{1}{\kappa^2\log(1/\kappa)})$ for stable offline algorithms and $\Theta(\frac{1}{\kappa^2})$ for online algorithms. Both thresholds lie far above the satisfiability scale $\Theta(\log(1/\kappa))$, as shown by Maillard~\cite{maillard2025}.

A dominating set $D$ of a graph $G$ is called $k$-limited if every vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality among all $k$-limited dominating sets of $G$ is the $k$-limited domination number, denoted by $\gamma_k^L(G)$. In this paper, we prove that the $1$-Limited Dominating Set problem is $\mathsf{NP}$-complete, answering an open question posed in the literature. We further show that, for every positive integer $k$, the $k$-Limited Dominating Set problem is $\mathsf{NP}$-complete even when restricted to planar graphs. In addition, we study the $k$-limited domination number of Sierpiński graphs. We determine the exact value of $\gamma_1^L(S(n,m))$ for all integers $n\ge 1$ and $m\ge 2$, and obtain the $k$-limited domination number for the planar family $S(n,3)$.

In the $q$-color online Ramsey game, Builder and Painter play on an infinite independent set of vertices. At each step, Builder draws an edge and Painter immediately assigns it one of $q$ colors. Builder aims to force a monochromatic copy of a fixed graph $H$. We prove that, for every $q \ge 2$ and $d \ge 1$, Builder can force a monochromatic copy of any $d$-degenerate graph $H$ while drawing a graph of degeneracy at most $d$. The bound $d$ is tight, and this resolves in the affirmative a problem of Conlon, Fox and Sudakov.

We continue the study of $k$-limited domination in graphs, a domination variant in which each vertex of a dominating set may dominate at most $k$ vertices outside the set. This concept extends classical domination by incorporating capacity constraints on dominating vertices. We improve the general lower bound for the $k$-limited domination number $\gamma_k^L(G)$ by employing the concept of $k$-capacitated domination. As a consequence, we characterize graphs satisfying $\gamma_k^L(G)=\lceil \frac{n}{k+1} \rceil$. We also refine the known upper bound for graphs with $k<\delta(G)$ using the $k$-limited packing number. Under this condition, we describe graphs attaining $\gamma_k^L(G)=n-k$. These results extend and unify previous investigations for the case $k=1$ and provide a complete characterization of graphs attaining the extreme values of the $k$-limited domination number under the considered assumptions.

A strong odd $k$-coloring of a graph $G$ is a proper $k$-coloring such that every color appearing in the neighborhood of a non-isolated vertex appears an odd number of times. The minimum $k$ for which $G$ admits a strong odd $k$-coloring is the strong odd chromatic number, denoted by $\chi_{\text{so}}(G)$, of $G$. Given a graph $G$ and an integer $k$, \textsc{strong odd $k$-colorability} problem asks whether $G$ admits a strong odd $k$-coloring. It is known that STRONG ODD $k$-COLORABILITY is NP-complete in general graphs. In this paper, we prove that the problem is NP-complete on perfect elimination bipartite graphs for $k\geq3$, which is a subclass of bipartite graphs. Furthermore, we show that $\chi_{\text{so}}(G)$ is inapproximable within a factor of $O(n^{\frac{1}{2}-\varepsilon})$ for every $\varepsilon>0$. On the positive side, we obtain a linear time algorithm to compute an optimal strong odd coloring for block graphs. From a parameterized perspective, we present an FPT algorithm for STRONG ODD $k$-COLORABILITY when parameterized by treewidth. Moreover, we show that the problem cannot be solved in time $(k-\varepsilon)^{\texttt{tw}}n^{O(1)}$ for every $k\geq3$ and $\varepsilon>0$ when parameterized by treewidth under SETH. Furthermore, we show that STRONG ODD $k$-COLORABILITY does not admit a polynomial kernel when parameterized by feedback vertex set. Lastly, we prove that STRONG ODD $k$-COLORABILITY is W[1]-hard when parameterized by clique-width.

A pentagonal geometry PENT(k,r) is a k-uniform, r-regular partial linear space with the property that the set of points not collinear with a given point x is itself a line in the geometry. The existence spectrum of these geometries is known; we begin the study of automorphisms of these structures in the case k=3. We determine the spectrum of pentagonal geometries PENT(3,r) with either a cyclic or reverse automorphism to be r = 1 or 10 (mod 12). In addition, we show that there are 584 pairwise non-isomorphic PENT(3,7)s and classify them by the orders of their automorphism groups.

A Neumaier graph is a non-complete edge-regular graph containing a regular clique; it is called strictly Neumaier if it is not strongly regular. In this paper we present a construction using finite rings that unifies several known results and yields three new families, each containing infinitely many strictly Neumaier graphs. We characterize when the resulting graphs are strongly regular. The family obtained from norm functions of finite field extensions includes the collinearity graphs of generalized quadrangles of \(*\)-Tits type arising from the classical hyperoval. We further show that, when the cyclotomic numbers involved have order at most seven, four types of rings suffice to obtain, up to isomorphism, all strictly Neumaier graphs arising from our construction. As a consequence of our results, each of these four types has now been used to construct Neumaier graphs.

In 1965, D. H. Lehmer conjectured that the permutations of every multiset admit an imperfect Hamiltonian traversal by adjacent swaps: a walk in the neighbor-swap graph that visits every word, with some words visited twice in order to reach a neighbor and return. The question is posed as an unsolved research problem in Knuth's Art of Computer Programming. Verhoeff (2017) chose the stutter words, in which every domino is a double, as the words to be reached this way, and reformulated the conjecture as the Hamiltonicity of the graph $N(S)$ on the non-stutter words, with two exceptional families — binary signatures with an odd multiplicity, and the permutations of $(2k,1,1)$ — that admit a Hamiltonian path but no cycle. This article proves the reformulated conjecture, and with it Lehmer's conjecture. The key structure is a partition of the words into hypercubes: the swaps inside dominoes turn each class of words with the same domino contents into a hypercube, and the stutters are exactly the $0$-dimensional classes. When every multiplicity is even, Hamiltonian cycles of the hypercubes are glued along a spanning tree, with no finite check. The case of exactly one odd multiplicity reduces to the all-even case and to a theorem of Stachowiak (1992), the one inherited Hamiltonicity input, which also settles two or more odd multiplicities. The only finite ingredients are two explicit cycles, of 28 and 84 words. Every construction is implemented in Python and checked against brute-force graphs, and the proof is formalized in Lean 4 over Mathlib.

Trisecting the dihedral angles of an $n$-simplex defines its Morley simplex. We study the original simplices for which this simplex is regular. A criterion in terms of the Gram matrix of the facet normals reduces the problem to a matrix equation. The derivative of the Morley map at the regular simplex has two explicit eigenvalues, both nonzero for $n\ge3$; thus the regular simplex is an isolated solution. We conjecture that in dimensions four and five a regular Morley simplex forces two hyperplane reflections interchanging disjoint pairs of vertices. We prove that this conclusion fails in every dimension $6\le n\le200$ and in every dimension $n=\binom k2-1$ with $k\ge9$: in these dimensions there are simplices with regular Morley simplex and no hyperplane reflection symmetry. The examples for $8\le n\le200$ have dihedral symmetry of order $2(n+1)$, while the infinite family is based on the Johnson scheme. In dimension four we give exact constructions of two nonregular examples, defined by irreducible polynomials of degrees $18$ and $8$ with Galois groups $S_{18}$ and $S_8$. Neither example is expressible by radicals. The computer assisted existence proofs use exact rational arithmetic and intervals with outward rounding.

We establish the coefficient sign patterns in the Third Borwein conjecture and the modulus-three Cubic Borwein conjecture. The analytic arguments apply for $n\ge1750$ and $n\ge500$, respectively. They combine exact dissections and positive coefficient identities near the boundary with saddle point estimates that preserve cancellation between primitive-root contributions. Paired Gaussian estimates remove the leading odd error, and explicit remainder bounds cover the complementary contours. The remaining finite intervals are checked by exact integer arithmetic. The Third finite verification through $1749$ is author-confirmed; the Cubic verification through $500$ is supported by two complete integer implementations and additional coefficient crosschecks.

The Berge conjecture [Proc. London Math. Soc., 1979] asserts that every bridgeless cubic graph can be covered by five perfect matchings. The 5-cycle double cover conjecture, proposed independently by Preissmann (1981) and Celmins (1984), asserts that every bridgeless graph admits five even subgraphs such that every edge belongs to exactly two of them. Hakobyan and Mkrtchyan [Ars Math. Contemp., 2019] proved that the 5-cycle double cover conjecture holds if and only if every bridgeless claw-free cubic graph can be covered by four perfect matchings. In this paper, we prove that the Berge conjecture holds for every bridgeless claw-free cubic graph $G$ and that $G$ has four perfect matchings covering at least $\frac{299}{315}|E(G)|$ edges.