Papers

We prove Norine's conjecture: every red–blue edge-coloring of the \(n\)-dimensional hypercube \(Q_n\), \(n\geq2\), in which antipodal edges have opposite colors contains a monochromatic path joining some vertex to its antipode. From a hypothetical counterexample we construct an antipodally equivariant, augmentation-preserving chain map from the cellular chains of the cubical boundary of a cube to subdivision-invariant polyhedral chains on a sphere of one lower dimension. A purely algebraic chain-level Borsuk–Ulam obstruction rules out this map.

Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge and a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for $n\geq 9$. Tait and Tobin (JCTB, 2017) proved the conjecture for sufficiently large order. In this paper, we completely resolved the Boots-Royle/Cao-Vince conjecture.

In incidence geometry, a triality is a symmetry cyclically exchanging triples of types of elements. Requiring geometries with trialities to satisfy standard regularity conditions makes their construction highly non trivial, and known examples are rare. In this paper, we present two infinite families of flag transitive, thin and residually connected geometries admitting trialities and no dualities with type preserving automorphism groups isomorphic to Sym(n) or Alt(n). We also develop general methods that extend to other settings. The residues of the two families are fundamentally different. In the first family, the maximal parabolics have constant size while in the other their size grows essentially linearly with the degree of the group.

Quasi-polynomials are ubiquitous in combinatorics and algebra, as they arise in a variety of enumeration problems. Because quasi-polynomials consist of constituent polynomials, their behavior is more subtle than for a single polynomial. In particular, unlike for a polynomial, it is possible for a quasi-polynomial defined on the positive integers to have infinitely many points at which it is decreasing. In this work, we characterize quasi-polynomials of degree $d$ and period dividing $p$ that are eventually nondecreasing, i.e., that have only finitely many values at which they decrease. We then give a detailed analysis of the space of eventually nondecreasing quasi-polynomials with fixed degree $d$ and period dividing a fixed $p$ such that the $h$-vector of the quasi-polynomial is nonnegative. Using this analysis, we determine the rate of growth of the number of such quasi-polynomials as a function of the sum of the $h$-vector entries for the $0$-th constituent polynomial.

We prove that $(k+1)$-uniform hypergraphs definable in an NIP strongly $k$-distal structure satisfy a homogeneous regularity lemma – they can be partitioned into a bounded number of simplicial complexes, most of which are homogeneous (meaning that the restriction of the hypergraph to the simplicial complex is either complete or empty). Furthermore, the parts of the partition can be chosen uniformly definably, and the size of the partition is polynomial in the reciprocal of the error parameter. This extends the homogeneous regularity lemma proven by Chernikov and Starchenko for hypergraphs definable in a distal structure. We prove this by introducing $k$-strong honest definitions and showing that an NIP structure is strongly $k$-distal if and only if every formula $\varphi(x_1, ..., x_k; y)$ has a $k$-strong honest definition. This extends the theory of strong honest definitions in distal structures to the higher-arity setting.

Special cases of finite cyclic semifields were first constructed by Hughes and Kleinfeld in 1960, and later by Sandler in 1962 and Knuth in 1965. The general construction of cyclic semifields was subsequently introduced by Petit in 1966, and later rediscovered from the perspective of irreducible semilinear transformations by Jha and Johnson in 1989. Since Sandler's foundational work in 1962, the complete determination of the autotopism groups of cyclic semifields and the full resolution of the isotopy problem for this family have remained long-standing open problems. The most significant advances in determining these autotopism groups are due to Dempwolff in 2011, who left open the case in which the field extension degree strictly divides the degree of the polynomial defining the semifield. In this paper, we provide a complete classification of cyclic semifields up to isotopy, together with the full determination of their autotopism groups, thereby closing the remaining cases left open by Dempwolff. Since cyclic semifields arise as a special instance of a broader family of maximum rank distance (MRD) codes constructed via skew polynomials, our methods also yield a complete classification of these MRD codes up to linear and semilinear equivalence over the prime field, together with an explicit description of their full automorphism groups.

For a totally ordered finite set $A$ with the total order $Q$, a poset $P=([d],\le_P)$, and a field $\mathbb{F}$, a tensor $T:A^d\longrightarrow \mathbb{F}$ is in $P$-echelon form if $r\le_{P}s$ in $P$ implies $a_r\le_Q a_s$ in every tuple $(a_1,\dots,a_d)$ in the support of $T$. We prove that if the Hasse diagram of $P$ has no isolated vertex, then tensors in $P$-echelon form with nonzero diagonal entries have full slice-rank. Our results extend and improve on recent results of Amanov and Yeliussizov.

We consider permutations $q$ in the symmetric group $S_n$ whose Verma multiplicities in the principal block of $\mathcal{O}(\mathfrak{sl}_n)$ satisfy $[M(p):L(q)] \geq 2$. We present a construction along with a diagrammatic visualization, showing how permutations in $S_n$ with this property generate a family of permutations in $S_{n+1}$ that share the same multiplicity property. While the method does not recover all such permutations in $S_{n+1}$, it systematically generates many new examples. In addition, we present all permutations in $S_5$ with $[M(\mathrm{id}):L(q)] \geq 2$ in a Bruhat diagram and describe all permutations in $S_6$ and $S_7$ with non-simple Verma multiplicities.

Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~$G=K_{(3,\,2^\beta)}$ are $e$-positive. We resolve this question and, together with their classification, characterize all $e$-positive complete multipartite graphs. Our main result is an explicit, manifestly nonnegative $e$-expansion of~$X_G$ whose coefficients are expressed in terms of the restricted-injection numbers. Our main idea is to derive a marker-variable coefficient-extraction formula for the $e$-coefficients of arbitrary complete multipartite graphs from the elementary–monomial Cauchy identity. For the particular graph~$G$, this formula reduces the proof to three coefficient families, which we evaluate using Dickson polynomials and recurrences for these numbers.

Factorization and decomposition of representative functions on a free monoid X * (generated by an alphabet X ) and with values in a ring A containing Q are equivalent to factorization and decomposition of their graphs (within the A-algebra of rational noncommutative series over X ) admitting linear representations (thanks to the Kleene-Schützenberger theorem). To factorize and to decompose effectively these graphs, we examine various products of noncommutative series (as concatenation, shuffle and its $\phi$deformations) and co-products such that, for A is a field K, their associated non graded commutative and co-noncommutative bialgebras of series are isomorphic to the Sweedler's dual of the graded noncommutative co-commutative bialgebras of polynomials having, for the concatenation, only Kleene stars of the planes as characters, or equivalently, only the planes are infinitesimal characters (thanks to a Ree's theorem like).

We study a local closure operation on the cover edges of a Boolean lattice: whenever the two lower edges or the two upper edges of a square face are present, all four edges of that square are added. We prove that every set of cover edges generating the full cover graph of $\mathcal{B}_n$ has cardinality at least $n$, and we classify all generators attaining this bound. For a graph $G$ on $[n]$, let $S_G=\{N_G(i)\to N_G(i)\cup\{i\}:i\in[n]\}$. Then $S_G$ diamond-generates the full cover graph if and only if $G$ is a cograph, and every minimum-cardinality generator arises uniquely in this way. Consequently, labeled minimum diamond-generating sets of $\mathcal{B}_n$ are in bijection with labeled cographs on $n$ vertices.

For every fixed pair of integers $k\ge 2$ and $q\ge 3$, we prove that every sufficiently large $k$-connected graph $G$ of order $n$ with minimum degree $\delta(G)\ge n/q$ contains a $k$-connected subgraph of every order $\ell\in\{2k,2k+1,\ldots,n\}$. In the case $k=2$, this confirms a conjecture of Liu and Ning~\cite{LiuNing}. The proof combines two constructions. First, we construct a small $k$-connected subgraph $D$ such that every vertex outside $D$ has at least $k$ neighbors in $D$. By successively adding the vertices outside $D$, we obtain $k$-connected subgraphs of every order from $|V(D)|$ to $n$. Second, an averaging argument on common neighborhoods, together with a complete bipartite construction, yields $k$-connected subgraphs of every order from $2k$ to $|V(D)|$. Together, the two constructions cover all orders from $2k$ to $n$. The lower endpoint $2k$ is best possible.

In this paper, we study a bootstrap percolation process on the finite triangular grid $\mathfrak{T}_n$ of side length $n$. We say that a subset $\eta$ of points in $\mathfrak{T}_n$ percolates if the final configuration, starting from $\eta$, is the whole grid $\mathfrak{T}_n$. A basis of size $n$ is then a subset of points of $\mathfrak{T}_n$ of minimum cardinality which percolates. In this paper, we first prove that the generating function counting bases satisfies an algebraic differential equation. Then, by analysing a modified version of this equation, we prove that the number $t_n$ of bases of size $n$ exhibits a stretched exponential asymptotic behaviour. More precisely, we show that $t_n \sim c n!e^{\sqrt{12n}}n^{5/12}$, for some constant $c>0$. These bases were recently shown by the second author to be in bijection with $3$-permutations avoiding the patterns $(12, 12)$ and $(231, 312)$, so this represents to our knowledge the first proven example of an asymptotic stretched exponential appearing in the study of pattern avoiding permutations.

Inspired by recent works on Diophantine tuples over finite fields, in this paper we consider Diophantine tuples related to primitive elements of finite fields. Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q^*=\mathbb{F}_q\setminus\{0\}$ be the multiplicative cyclic group of all non-zero elements over $\mathbb{F}_q$. An element $g\in\mathbb{F}_q$ is called primitive if $g$ generates the group $\mathbb{F}_q^*$. A set $\{x_1,x_2,\cdots,x_m\}\subseteq\mathbb{F}_q^*$ of $m$ elements is said to be a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ if $x_ix_j+1$ is primitive for any $1\le i\le j\le m$. Let $N_m$ denote the number of $\mathcal{P}$-Diophantine tuples over $\mathbb{F}_q$. Then we obtain the asymptotic formula $$m!\cdot N_m=\left(\frac{\varphi(q-1)}{q-1}\right)^{m(m+1)/2}q^m+O_{m,r}\left(q^{m-\frac{1}{2}+r}\right),$$ where $\varphi(\cdot)$ is the Euler totient function and $r\in(0, 1/2)$ is an arbitrary real number. Moreover, we prove that there exists a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ whenever $q\ge \exp(\exp(m(m+1)))$.

The global and local $\mathcal{G}$-covering number $c_{\mathrm{g}}^{\mathcal{G}}(H)$ and $c_{\mathrm{l}}^{\mathcal{G}}(H)$ encode how well the edges of a graph $H$ can be covered with graphs from a graph class $\mathcal{G}$: in the global setting, we minimize the number of graphs from $\mathcal{G}$ required, in the local setting how often a vertex is hit by the graphs of the cover. Within this work we consider for $\mathcal{G}$ the graph classes $\mathcal{B}$ of all bipartite and $\mathcal{B}_{\mathrm{c}}$ of all complete bipartite graphs. We give a tight lower bound on $c_{\mathrm{l}}^{\mathcal{B}}(H)$ in terms of the fractional chromatic number of $H$, thereby giving a local analogue of a result by Harary, Hsu and Miller. Answering a question by Fishburn and Hammer, we show that it is NP-hard to determine $c_{\mathrm{l}}^{\mathcal{B}_{\mathrm{c}}}(H)$. Further, we provide a finite and monotone graph class $\mathcal{G}$ such that $c_{\mathrm{g}}^{\mathcal{G}}(H)$ can be computed in constant time for every graph $H$ while determining $c_{\mathrm{l}}^{\mathcal{G}}(H)$ is NP-hard. This yields a natural example to a question raised by Knauer and Ueckerdt.

Circulant digraphs are Cayley digraphs over finite cyclic groups and constitute a fundamental class of objects in algebraic graph theory. Extending the classification of locally-primitive circulant graphs \cite{JZ-2026}, we completely determine all locally-quasiprimitive circulant digraphs. Our main theorem shows that a connected locally-quasiprimitive circulant digraph is isomorphic to one of the following: the complete graph \(\K_n\), the complete bipartite graph \(\K_{n/2,n/2}\), the graph \(\K_{n/2,n/2}-\frac{n}{2}\K_2\) (with \(n/2\) odd), the cycle \(\C_n\), the directed cycle \(\vec \C_n\), a normal circulant digraph of prime valency, the lexicographic product \(\vec \C_m[\overline{\K_b}]\), or the tensor product \(\vec \C_m\times \K_b\) with \(\gcd(m,b)=1\).

We prove that monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial. The proof combines Demazure operators with vector partition functions and, in the Schubert case, P.~Magyar's orthodontic formula. These coefficient results extend to finite products. Using M.~Watanabe's Schubert duality, we deduce that stretched Schubert structure constants are eventually polynomial, proving a conjecture of I.~Pak and Z.~Slonim. For key polynomials, this resolves the polynomiality part of a conjecture of P.~Alexandersson and E.~Alhajjar.

The rearrangement number $\mathfrak{rr}$ is the least cardinality of a collection of permutations of $\omega$ such that every conditionally convergent real series is disrupted by some permutation in the collection. Blass, Brendle, Brian, Hamkins, Hardy, and Larson proved that $\max\{\operatorname{cov}(\mathcal N),\mathfrak b\}\leq\mathfrak{rr}\leq\operatorname{non}(\mathcal M)$ and asked whether $\mathfrak{rr}=\operatorname{non}(\mathcal M)$. We prove in ZFC that $\mathfrak{rr}=\operatorname{non}(\mathcal M)$.

We study the lifting problem for mod $2$ characteristic maps over simplicial $3$-spheres. Using a bad-block partition of the universal complex $X(\mathbb{Z}_2^4)$, we prove an avoidance criterion for liftability. We show that every simplicial $3$-sphere with at most $20$ vertices has the toric lifting property. We also obtain image-size and join-type results, and prove sharpness of the image-size bound in the universal-complex sense.

For a finite set $A\subset\mathbb{Z}$, write $hA$ for its $h$-fold sumset, and let \[ R(h,k)=\{|hA|:A\subset\mathbb{Z},\ |A|=k\}. \] We determine the part of $R(h,4)$ lying between $4h+2$ and $6h-4$: for $h=4$ the only value is $5h-1$, while for $h\geq 5$ the only values are $5h-1$ and $5h+1$. This proves Rajagopal's conjectured gap $5h\notin R(h,4)$ for every $h\geq 4$. For $h\geq 6$, it also yields the new missing interval $[5h+2,6h-4]$, which lies outside Rajagopal's general excluded set. Lev's lower bound for the successive growth of multiple sumsets reduces the problem to normalized sets of affine diameter five, of which there are only six. Reflection and four elementary exact sumset computations finish the classification.

In combinatorial game theory, the choice of play convention is a fundamental aspect of the theory. The two most widely studied conventions are normal play, in which the player who makes the last move wins, and misère play, in which the player who makes the last move loses. Another well-known variant is play with a single shared pass. In such games, at most one pass may be used in total during the game: once the pass has been used, neither player may pass thereafter. Moreover, once a terminal position has been reached, passing is no longer allowed. Recall that, for misère play, one sometimes defines SG values (or Sprague-Grundy values) in the same recursive manner as in normal play but assigns the terminal position the value 1. Also, SG values can be considered for normal-play games with a pass. In this work, we propose transformations that allow both misère play and play with a pass to be treated as normal-play games. As a consequence, we show that there is a certain duality between a generalization of misère play and a generalization of play with a pass.

Regularity and depth of binomial ideals generated by adjacent $2$-minors together with those arising from finite lattices are studied.

Qualitative independence hypergraphs provide a useful combinatorial framework for analyzing the existence and structure of covering arrays. In this work, we study the uniform and \emph{almost-uniform $3$-qualitative independence hypergraphs} $3\text{-}UQI(n,2)$ and $3\text{-}AUQI(n,2)$, and establish a structural correspondence between these families and merged Johnson graphs, with emphasis on their cores. Focusing on the smallest unresolved instance, $3\text{-}QI(8,2)$, we classify all of its strongly independent sets and determine its strong independence number. Using this, along with its strong chromatic number and the size of the largest $3$-clique, we show that $3\text{-}QI(8,2)$ is a core. For $n>8$, we further identify sufficient conditions under which $3\text{-}UQI(n,2)$ and $3\text{-}AUQI(n,2)$ are cores.

We determine the strong independence number of the qualitative independence hypergraph, $3\text{-}QI(11, 2)$, using a technique that involves considering its vertices as subsets of $\{1,2, \ldots, 11\}$ and assessing them as intersecting set systems. This gives the maximum size of colour classes in any strong colouring and thus, a lower bound on the strong chromatic number of $3\text{-}QI(11,2)$. We leverage this bound along with an upper bound of the strong chromatic number of $3\text{-}QI(10,2)$, to consequently, establish that the covering array number of $3\text{-}QI(11, 2)$, $CAN(3\text{-}QI(11,2),2) = 11$ and give a sufficient condition for a hypergraph $H$ to have $CAN(H, 2)=11$.

Two families of polytopes underlie the combinatorics of associative operations. Associahedra and multiplihedra respectively capture the information in the operation itself and in the morphisms that respect that operation. The first applications of these polytopes, from Stasheff, were for modeling homotopy associative spaces and their homotopy homomorphisms. Later, lax and weak higher categories used both as the shapes of commuting diagrams. More recently, the associahedra have been generalized to versions based on graphs, and then to acyclic versions based on posets: the acyclonestohedra. Meanwhile, the associated multiplihedra have also been generalized to graph associahedra and generalized permutohedra. Here we complete that picture: we use the graph multiplihedra to define and realize new acyclic poset multiplihedra for the acyclic poset associahedra. Quotients of these are also studied, concluding with the conjectured interval polytopes of the order polytopes of posets.

Barry introduced a central transform of integer sequences and proposed four conjectures concerning the Hankel transforms of central transform of four rational families. We prove these four conjectures. The proofs are unified within a common algebraic framework: we interpret the Hankel determinants as Gram determinants and use a basis of shifted monic Chebyshev polynomials to reveal the finite-band structure of the associated Gram matrices.

For a (molecular) graph $G$ and any real number $\alpha\ne 0$ , the zero-order general Randić index , denote by $^0R_\alpha$, is defined by the following equation: \begin{align*} {^0R_\alpha} (G) =\sum_{v\in G}d_G (v) ^{\alpha} (\alpha \in \mathbb{R}-\left\{0\right\}) . \end{align*} The deficiency of $G$, denoted by $def(G)$, is equal to the cardinality of vertices which are not covered by a maximum matching in $G$. A graph G is called $l$-deficient if $def(G)\le l$. In this paper, we use this index to give sufficient conditions for a connected graph, bipartite graph and a balanced bipartite graph $G$ to satisfy the $l$-deficient property, and show that none of these conditions can be dropped. We will also use these results to enhance and generalise the results that already obtained by M. An and K. C. Das in 2018 and G. Su et al. in 2022.

In this article we establish global a priori estimates on the solution of a generic rough differential equation driven by an $\alpha$-Hölder path covering the full range of regularity $\alpha\in(0,1)$, under the hypothesis of Lipschitz continuity (but neither boundedness nor coercivity) of specific combinations of the coefficients and their derivatives, called elementary differentials. Our main tool is a closed formula for the remainder of a generalised Taylor expansion in terms of products of gradients of the elementary differentials, which allows to take advantage of cancellations in an optimal way.

For a graph $G$ and an integer $d \geq 0$, let $\chi^d(G)$ denote the $d$-defective chromatic number, and let $G \boxtimes K_{d+1}$ be the $(d+1)$-fold clique blowup of $G$. Norin and Steiner disproved the conjecture $\chi(G) = \chi^d(G \boxtimes K_{d+1})$ of Guo, Kang and Zwaneveld by exhibiting, for infinitely many $d$, graphs with $\chi(G) \geq (30/29) \chi^d(G \boxtimes K_{d+1})$, and they proved the universal upper bound $\chi(G) \leq 2 \chi^d(G \boxtimes K_{d+1})$. Writing $C_d = \sup_G \chi(G)/\chi^d(G \boxtimes K_{d+1})$ and $C^* = \sup_d C_d$, their results give $C^* \in [30/29, 2]$. We improve the lower bound: we exhibit an explicit 40-vertex graph $W$ with $\chi(W) = 11$ and $\chi^2(W \boxtimes K_3) = 10$, so that $C^* \geq C_2 \geq 11/10 > 30/29$, already at the smallest defect for which such a separation is possible, namely $d = 2$. All parameters are established by the proofs; the only computer-assisted input, the non-list-colourability of a certain 30-vertex, 10-colour list instance $(B,L)$, is certified by an independently checkable DRAT refutation.

We present a corrected version of a theorem from the paper "N-free posets and orthomodularity" published in Order 43(1) (2026).