Papers

For a finite group $G$, $N_\infty$ operads encode collections of norm maps, and by work of Blumberg–Hill and Rubin their homotopy category is equivalent to the poset of $G$–transfer systems on the subgroup lattice of $G$. In \cite{ABB+25} the authors defined the width $w(G)$ as the minimal size of a generating set for the complete $G$–transfer system and identified it with the number of conjugacy classes of proper meet irreducible subgroups of $G$, and the complexity $c(G)$ as the maximum, over all transfer systems $T$, of the size of a minimal generating set for $T$. We compute $w(G)$ for the semidihedral groups $\SD_{2^n}$ ($n\ge 4$) and the affine Frobenius groups $\AGL(1,p^n)\cong \mathbb{F}_{p^n}\rtimes \mathbb{F}_{p^n}^\times$, extending existing calculations and highlighting how subgroup lattice structure governs equivariant multiplicative complexity. We also compute $c(D_{p^n})$ for dihedral groups of order $2p^n$ with $p$ an odd prime, establishing $c(D_{p^n})=\lfloor 3n/2\rfloor+1$, and derive the lower bound $c(\SD_{2^n})\ge\lfloor 5(n-1)/2\rfloor$.

Let $D_d(m)$ be the Cartesian product of $d$ positively oriented directed cycles of length $m$. We prove that $D_d(m)$ decomposes into $d$ directed Hamilton cycles for every $m\ge3$ and $d\ge2$. The construction proceeds by splitting coordinate directions in directed multitori. An integer selection theorem supplies unit voltages compatible with the prescribed arc multiplicities; at even modulus, the decisive condition is the parity of each incidence component. For even $m$ and odd $d\ge7$, we satisfy this condition with a factorization having one additional cycle. A relative lifting theorem preserves the first-return data of a three-colour recolouring through successive coordinate splits, after which the recolouring removes the additional cycle on a set whose size is independent of the dimension. Explicit constructions complete the low-dimensional cases. The theorem yields Hamilton decompositions of Cartesian products of equal-order, equal-degree Hamilton-decomposable digraphs, of abelian Cayley digraphs whose generators partition into module bases, and of a family of height-stretched directed tori.

Let $G$ be a connected graph with vertex set $V(G)$, and denote by $d_G(u,v)$ the distance from $u$ to $v$ in $G$, for any $u,v \in V(G)$. The average distance of an $n$-vertex connected graph $G$, denoted by $\mu(G)$, is defined to be the average of all distances between all pairs of vertices in $G$, i.e., $\mu (G) = \binom{n}{2}^{-1} \sum_{\{u,v\} \subset V(G)}d_G(u,v)$. The problem of finding a spanning tree of minimum average distance is known to be NP-hard, so establishing an upper bound for the minimum average distance among all spanning trees is of particular interest. Mukwembi (J. Graph Theory, 2014) showed that if $G$ is a connected graph of order $n$ with independence number $\alpha$, where $n > 2 \alpha - 1$, then $G$ has a spanning tree $T$ such that $\mu(T) \le \alpha + 2$. In this paper, we first improve the upper bound to $\mu(T) < \alpha + 1$ for $\alpha \ge 1$, and then we find the bound could be further improved when $\alpha$ becomes larger, so a better upper bound \[ \mu(T) < \left\{ \begin{array}{ll} \alpha+1 & \hbox{if } 1\le\alpha\le 6,\\ \alpha+\frac12+\frac{4(\alpha-1)}{\alpha^2} & \hbox{if } \alpha\ge 7, \end{array} \right. \] is established later. In the end, we give a remark to indicate our new upper bound is best possible in the sense of asymptotics (when $n$ and $\alpha$ are large enough).

We study Keisler measures in strongly n-distal NIP theories, generalizing some results of Simon and Chernikov-Starchenko for distal theories and addressing some questions of Walker. In particular, we establish a hypergraph version of the distal regularity lemma, compact domination for definable fsg groups, and demonstrate that the strong n-distality hierarchy is strict among stable theories using a connection to Poizat's total triviality of forking. We also show that infinite strongly n-distal NIP fields have characteristic 0 using a discrepancy result of Babai-Hayes-Kimmel from multiparty communication complexity.

Motivated by Wegner's conjecture on squares of planar graphs, Thomassen conjectured that every 3-connected cubic graph on at least eight vertices admits a red-blue vertex coloring in which the blue subgraph has maximum degree at most 1, while the red subgraph has minimum degree at least 1 and contains no $P_4$. Such colorings are now called crumby colorings. Although this conjecture was disproved in general by Bellitto, Klimošová, Merker, Witkowski and Yuditsky, positive results of Barát, Blázsik and Damásdi led them, in the same subcubic setting, to conjecture that every $K_4$-minor-free graph admits a crumby coloring. We disprove this conjecture with a connected subcubic partial 2-tree on 18 vertices. We also disprove its natural 2-connected version with a 2-connected subcubic partial 2-tree on 40 vertices with no crumby coloring. Consequently, the obstruction to crumby colorability already occurs within treewidth two, even under 2-connectivity.

We construct sequences $\{a_n\}_{n\in\mathbb{N}}\in\{-1,1\}^{\mathbb{N}}$ with small values of signed harmonic sums \[ \sum_{n\in\mathcal{A}\cap[1,N]}\frac{a_n}{n}, \] for any reasonably dense subsets $\mathcal{A}\subset\mathbb{N}.$ We apply these methods to further construct completely multiplicative functions $f:\mathbb{N}\to\{-1,1\}$ with unusually small logarithmic partial sums, that is, \[ \sum_{n \leq N}\frac{f(n)}{n} \ll \exp\left(-c_0 \frac{N^{1/3}}{(\log N)^{1/3}} \right) \] holds for infinitely many $N\to\infty$. The proofs combine careful analysis of the small-scale distribution of random harmonic sums over subsets of $\mathbb{N}$, together with deterministic inductive arguments inspired by the “anatomy" of integers.

Coset geometries are incidence geometries constructed from a group $G$ and a system of subgroups $(G_i)_{i \in I}$ of subgroups of $G$. For any algebraic group operation, it is then natural to wonder whether it can be extended to the framework of coset geometries. This has been achieved in the case of the halving (\cite{halving}) and in the case of free (amalgamated) products, HNN-extensions, and semi-direct products (\cite{piedade2025group}). In this article, we explore more deeply two operations related to semi-direct products: the twisting and the wreath product. We show that these operations extend to coset geometries in such a way that they preserve key properties, such as flag-transitivity, residual-connectedness and being thin. In particular, we can apply twistings and wreath products to polytopes and hypertopes. Doing so, we show that there exists regular polytopes and hypertopes for almost-simple group with socle a sporadic simple group.

We present CombOL (Combinatorial Objects Library), an open-source library for the enumeration and Boltzmann sampling of combinatorial classes. Classes can be specified by a concise string syntax, and may depend on an arbitrary number of parameters. CombOL automatically derives the associated generating functions, enabling the generation of counting sequences and the compilation of Boltzmann samplers. The library supports exact and approximate-size Boltzmann rejection sampling with automatic parameter tuning to target specific sizes. In addition to implementing established methods, CombOL contributes a novel early-rejection scheme, as well as guaranteed statistical correctness by dynamically increasing the numerical precision, eliminating bias due to floating-point rounding errors. Through the Python interface, sampled structures can be mapped to application-specific objects, enabling direct sampling of domain objects such as graphs, chemical structure representations, or other complex data types. CombOL is available from PyPI as 'combol' (pypi.org/project/combol). The source code is available at gitlab.com/casbjorn/combol.

Ilse Fischer and the second author introduced in [Algebr. Comb. 7 (2024), no. 5, 1319-1345] a two parameter family of polynomials defined as sums over totally symmetric plane partitions and connected to alternating sign matrices and cyclically symmetric lozenge tilings of a hexagon with a triangular hole. In this paper we present several determinantal formulae leading to new lattice path models and a novel family of tableaux. The later illustrates that the polynomials of our interest can be thought of as generalisations of the three dual Littlewood identities.

A whirling knight's tour is a Hamiltonian cycle in the digraph of counter-clockwise knight steps about the centre of an $n \times n$ board; its coil count $c$ is the winding number around the centre. We prove that no such tour with $c = n/2$ exists when $n \equiv 4 \pmod 8$ ($n \ge 4$) or $n \equiv 6 \pmod 8$ ($n \ge 6$), settling a conjecture of Beluhov. For each residue class we exhibit a closed-form Farkas certificate for infeasibility of a cycle-cover LP relaxation; the two certificates are structurally distinct.

The proper commuting graph $\mathcal{C}^{**}(G)$ of a finite group $G$ is the simple graph whose vertices are the noncentral elements of $G$ and two distinct vertices are adjacent if they commute. In this paper, we study the domination number and total domination number of proper commuting graphs of finite groups. We first obtain general bounds for the domination number of proper commuting graphs. For finite nilpotent groups, we exploit a strong product decomposition of commuting graphs to derive exact formulas for the domination number. We further determine the exact domination number and total domination number for proper commuting graphs of several well-known families of finite groups, connecting with the centralizers of those groups.

Let $G$ be a finite graph on $[n]:=\{1, \ldots, n\}$ and $\kappa(G)$ its vertex connectivity. Let $S=K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph $G^c$ of $G$. It is a classical result that ${\rm depth} S/I(G^c) \leq \kappa(G) + 1$. We give a sharp lower bound of ${\rm depth} S/I(G^c)$ in terms of $n$ and $\kappa(G)$. Furthermore, a sharp lower bound of ${\rm depth} S/I(G^c)^2$ as well as that of ${\rm depth} S/I(G^c)^{(2)}$ in terms of $n$ and $\kappa(G)$ is given.

A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a cornerstone of modern matching theory. Determining whether a given graph is matching covered is a fundamental problem in structural graph theory. Lucchesi et al. [SIAM J. Discrete Math., 2018] showed that a connected graph $G$ is matching covered if and only if every barrier of $G$ is a stable set. In this paper, we completely characterize the extremal graphs that maximize the size or the spectral radius among all non-matching-covered graphs. For $a \leq b$ and $b \geq 2,$ Hao and Li [Electron. J. Combin., 2024] investigated the extremal problems on $[a,b]$-factor graphs: If $G$ contains no $[a,b]$-factors, then $e(G)\leq \binom{n-1}{2}+a-1$ with equality if and only if $G\cong H_{n,a},$ where $H_{n,a} = K_{a-1} \vee (K_{n-a} \cup K_1).$ Moreover, if $G$ contains no $[a,b]$-factors, then $\rho(G)\leq \rho(H_{n,a})$ with equality if and only if $G \cong H_{n,a}.$ Judging from the structral characterization, non-$[a,b]$-covered graphs exhibit highly complex structures, making the associated extremal problems significantly challenging. To overcome this, we develop a novel minimum-degree forcing technique. Combining this technique and spectral-structural analysis, we in this paper provide complete characterizations of the extremal graphs that maximize the size or the spectral radius within the set of non-$[a,b]$-covered graphs. An intriguing phenomenon revealed by our results is that $H_{n,a}$ remains both the size-extremal graph and the spectral extremal graph for this larger set of non-$[a,b]$-covered graphs. Consequently, our results strengthen the results of Hao-Li.

This paper studies uniform mixing in continuous-time quantum walks. We show that for some unitary signing $\sigma$, the complete graph $K^\sigma_n$ has probabilistic uniform mixing. In contrast, Ahmadi \etal (2003) proved that no complete graph has uniform mixing except for $K_2$, $K_3$, and $K_4$. Our technique is based on a stopping rule for quantum walks which reduces global to local uniform mixing. As a corollary, we found an orientation of $H(n,4)$ that mixes to uniform faster than any other Hamming graphs, which improves a result of Godsil and Zhan (2019). We also show that there are infinite families of oriented circulants with average uniform mixing. This is a chiral violation of a No-Go theorem due to Godsil (2013) which states that no graph has average uniform mixing except for $K_2$.

Let $e(n,s)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. In 1968, answering a question of Erdős, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integers $m,s\ge 1$. Half a century later, Frankl and Kupavskii determined $e(s(m+1)-\ell, s)$ for $\ell \leq \frac{s-3}{m+3}$. They showed that the corresponding extremal example is closely connected with the extremal example for the Erdős Matching Conjecture, and conjectured that the same remains true for all $\ell \leq s/2$. In this paper, we prove an approximate version of their conjecture for $s\ge s_0(m)$.

For the OEIS sequence A176677, defined by the quadratic convolution recurrence $a(0) = a(1) = 1$ and $a(n+1) = \sum_{p=0}^n a(p) a(n-p) - 1$ for $n \ge 1$, R.~J.~Mathar contributed in March 2016 the conjectured order-4 P-recursive recurrence \[ (n+1)\,a(n) + 2(-3n+1)\,a(n-1) + (9n-13)\,a(n-2) - 4\,a(n-3) + 4(-n+4)\,a(n-4) = 0. \] We give a short proof. The convolution recurrence translates directly into the algebraic equation $z(1-z) G(z)^2 - (1-z) G(z) + (1 - z - z^2) = 0$ for the ordinary generating function $G(z)$, and Mathar's recurrence then drops out as the coefficient form of a 1st-order linear inhomogeneous ODE $q_0(z) G(z) + q_1(z) G'(z) = R(z)$ that we verify by polynomial division modulo the algebraic equation. The polynomial $q_1(z)$ admits the factorization $q_1(z) = -z(z-1)(2z-1)(2z^2 + 3z - 1)$, whose roots are exactly the singularities of $G$. Deutsch's combinatorial interpretation (Motzkin paths of length $n-1$ with two-coloured level-zero horizontal steps) is preserved.

The $n$-th Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by binary strings with no two consecutive ones. We determine $\pi(\Gamma_n) = 2^n$ for $n \le 6$, so the pebbling number of $\Gamma_n$ equals that of the ambient hypercube $Q_n$ despite $\Gamma_n$ having far fewer vertices. The lower bound is a standard potential argument. For the upper bound, the Weight Function Lemma yields $2^n+1$ – one too many – so we close the gap by exhaustive MILP verification. We conjecture $\pi(\Gamma_n) = 2^n$ for all $n$.

We introduce a unified theoretical framework for the rigorous analysis and systematic construction of deep neural networks (DNNs). This framework addresses a gap in existing theory by explicitly modeling the structure of tensor operations – lower level information that is often abstracted. Our framework enables two novel objectives: (1) analysis of the evolution of architectural complexity over deep learning history, and (2) automatic construction of novel architectures based on new types of tensor operations. Our study of DNNs introduced over the past 40 years reveals a connection between groundbreaking architectures and increases in different types of architectural complexity. Moreover, we identify several large classes of higher complexity architectures that have not yet been explored. We then collect a dataset of 3,000+ higher complexity architectures, which we publicly release at: https://github.com/combinatoriallabs/ArchitecturalComplexity.

The goal of the paper is to present two simple proofs of the Lagrange Inversion Formula for formal power series. Both proofs are non-external in the sense that they use concepts that do not go beyond the scope of basic tools of formal power series analysis.

Quantile shares, introduced by Babichenko, Feldman, Holzman, and Narayan [STOC 2024], offer an ordinal, self-maximizing, and interpretable benchmark for fair division of indivisible goods, but their universal feasibility is known only conditional on the rainbow Erdős matching conjecture (EMC). Specifically, Babichenko et al. showed that assuming the rainbow EMC in the near-perfect matching regime, the $(1/2e)$-quantile share is universally feasible. In contrast, a simple argument shows that the $q$-quantile share can be infeasible for any $q > 1/e$. We introduce a one-parameter refinement of quantile shares, the $c$-thinned quantile share, obtained by thinning the inclusion probability in the random benchmark bundle by a factor of $c$ for a fixed constant $c\in(0,1]$. Our main result is that there exists a universal constant $c >0$ for which the $c$-thinned $e^{-c}$-quantile share is unconditionally universally feasible; this is best possible in the sense that for any $c \in (0,1]$, the $c$-thinned $q$-quantile share can be infeasible for any $q > e^{-c}$. Prior to this work, the only nontrivial share known to be universally feasible was Feige's residual maximin share. The thinning viewpoint also lets us remove the factor-two loss in the conditional result for the original quantile share: assuming the rainbow EMC, the $(1/e)$-quantile share is universally feasible.

Classroom dynamics depend on various elements that influence teaching performance and learning activities. A key challenge is to determine the most effective seating plan, where students will seat in a specific classroom setting to achieve the best learning environment. This paper introduces the Student Seat Allocation Problem (SSAP) for strategically organizing student seating in traditional classrooms to minimize interpersonal conflicts. We propose a mathematical model and an Iterated Local Search (ILS) heuristic to solve the SSAP. Computational experiments demonstrated that ILS outperformed in more complex scenarios when compared to the results obtained by a commercial solver on the introduced mathematical model. ILS was particularly efficient in real and artificial instances that exhibited a higher number of conflicts.

We give a concrete combinatorial interpretation of the coefficients of the Kazhdan-Lusztig polynomials of Dowling geometries, a family of matroids which generalizes braid matroids of types A and B. Furthermore, we interpret the coefficients of the equivariant Kazhdan-Lusztig polynomials and the equivariant Z-polynomials of Dowling geometries associated to non-trivial groups with respect to their full automorphism groups.

Team captains Alice and Bob divide up $2m$ footballers, each reduced to a real-valued score, into two teams of $m$ footballers each. On each turn, one captain plays picker, and the other chooser: the picker names a footballer yet to be selected, and the chooser decides which captain's team receives that footballer. Alice starts as picker, Bob as chooser, and roles alternate. The game ends as soon as either captain has a full team of $m$ footballers, at which point the other team receives all remaining footballers. The team with the larger sum of scores wins. Settling a problem raised by Eccles in 2015, we show that Alice cannot win.

We study the chromatic number of graphs that exclude a clique as a strong odd immersion and have independence number two. Given a graph $G$ and $t\in\mathbb{Z}^+$, we prove that if $\alpha(G)\leq 2$ and $G$ has no strong odd $K_t$-immersion, then $\chi(G)\leq \lceil \frac{3(t-1)}{2}\rceil$.

We develop a saddle-point theory for acyclic orientations and negative chromatic evaluations of complete multipartite graphs, with applications to OEIS A267383, A372326, A372084, A372395, and A370613. The main tool is an exact Gamma-type integral representation for acyclic orientation counts and its Gamma-weighted extension to the negative chromatic axis. We prove Kotesovec's fixed-column conjecture for A267383 for arbitrary fixed numbers of parts, give the corresponding fixed-p Tutte-axis asymptotics, develop an analytic-combinatorics-in-several-variables framework for chromatic evaluations of fixed graph blow-ups, and give unconditional fixed-base families reducible to balanced Turan graphs. In the product regimes we prove fixed part-size and finite-profile expansions, and for equal-size parts we obtain an all-order expansion throughout every fixed polynomial window, including explicit corrections through the cubic scale. Finally, we prove logarithmic asymptotics for the partition-sum sequences A372395 and A370613 via a quadratic-energy partition model, a growing-window comparison for the Stirling-transform factors, and a random-permutation far-tail bound.

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties; however, the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, ${\{P_t,K_{\ell,\ell}\}}$-free graphs have bounded tree-independence number. We prove this conjecture for ${t=5}$ by showing that every ${\{P_5,K_{\ell,\ell}\}}$-free graph has tree-independence number at most ${4\ell-4}$. We also obtain related bounds for the weaker parameter of $\alpha$-degeneracy and answer a question of Hilaire, Milanič, and Vasić whether tree-independence number of ${\{P_5,K_{\ell,\ell}\}}$-free graphs exceeds $\ell$ by at most an additive constant.

Hypergraphic polytopes $\Delta_{\mathbb{H}}$ arise as Minkowski sums of simplices indexed by the hyperedges of a hypergraph $\mathbb{H}$. Orienting the $1$-skeleton of such a polytope by a certain generic linear functional gives rise to the hypergraphic poset $P_{\mathbb{H}}$. Hypergraphic posets include the weak order for the permutahedron and the Tamari lattice for the associahedron. This motivates the problem of determining when $P_{\mathbb{H}}$ is a lattice. In this paper, we give a complete lattice characterization for cyclic interval hypergraphs, extending the result of Bergeron and Pilaud for interval hypergraphs, and the result of Adenbaum et al. for the complete cyclic interval hypergraph.

For a graph $G$ with vertex set $V(G)$ and a positive integer $i$, an $i$-packing in $G$ is a subset $X$ of $V(G)$ such that the distance between any two distinct vertices of $X$ is greater than $i$. The packing chromatic number of $G$, denoted by $\chi_{\rho}(G)$, is the smallest positive integer $k$ for which there exists a partition $X_1, X_2, \ldots, X_k$ of $V(G)$ such that $X_i$ is an $i$-packing in $G$ for every $i \in [k]$. A graph $G$ is called $\chi_\rho$-critical if $\chi_\rho(H) < \chi_\rho(G)$ holds for every proper subgraph $H$ of $G$. In this paper, we provide a structural characterization of $\chi_{\rho}$-critical graphs with radius $1$, and completely determine the $\chi_{\rho}$-critical cactus graphs with radius $2$ and diameter $2$ or $3$.

On a finite weighted graph, the dimer model is a probability measure on its dimer covers, that assigns to any cover a probability proportional to the product of the weights of its edges. For planar bipartite graphs, dimer correlations are encoded by the inverse of the so-called Kasteleyn matrix; for a large graph, typically taken as a finite domain in a periodic graph, this inverse matrix is known explicitly only for a handful of examples. In all previously known examples, the Newton polygon – a convex lattice polygon that classifies periodic graphs up to local moves – is either a triangle or a quadrilateral. Our main results are the following. For any (minimal) periodic planar bipartite graph, we construct an $(n-3)$-dimensional family of finite subgraphs for which we obtain an explicit inverse Kasteleyn matrix; here $n$ is the number of sides of the Newton polygon. Their boundaries are formed by zig-zag paths and their overall shape is reminiscent of an astroid; we call them astroidal zig-zag graphs (AZ graphs). If the Newton polygon is the unit square then the corresponding AZ graph is the celebrated Aztec diamond with its size as the parameter. Our inverse Kasteleyn matrices are given by a double contour integral on the corresponding spectral curve for any Fock weighting of the graph. This includes, in particular, all periodic weightings. For periodic weightings, we asymptotically analyze the resulting inverse Kasteleyn matrices. We establish a phase separation in large AZ graphs into asymptotically frozen, rough (liquid), and smooth (gaseous) regions, and obtain an explicit parametrization of the `arctic curve'. We also compute the deterministic limit of the height function, known as the limit shape, and prove the convergence of the local dimer correlations to the translation-invariant Gibbs measure of the slope predicted by the limit shape.

We introduce the degree filtration on the discrete cubical chain complex of a graph, defined in terms of the maximal injective dimension of the facets of singular $n$-cubes, and study the degree spectral sequence which arises from this filtration. This spectral sequence interpolates between the discrete cubical homology of a graph $H_n(G)$ and the injective homology $H_n^{inj}(G)$, a variant of the discrete cubical homology based on injective singular cubes. Building on the work of Greene and the first author, we introduce the combinatorial condition of quasimonophobicity on graphs, and show quasimonophobicity implies both the vanishing of the degree spectral sequence in certain bidegrees, and implies $H_n^{inj}(G)$ is isomorphic to the homology of the CW complex obtained by “filling in” subcubes of the graph. These results are applied to compute $H_2(G_n^{sph})$ for the Greene sphere graphs $G^{sph}_n$.