Papers

Let $\Gamma=(V,E)$ be a graph. The disjunctive domination number of $\Gamma$ is the minimum cardinality of a set $S\subseteq V$ such that every vertex not in $S$ is adjacent to a vertex of $S$, or has at least two vertices in $S$ at distance $2$ from it. In this paper, we give bounds for the disjunctive domination numbers of the torus grid graphs $C_m\Box C_n$, and determine the disjunctive domination numbers of $C_3\Box C_n$, $C_4\Box C_{n}$ and $C_8\Box C_{4n}$.

The wakeup problem in distributed computing asks for a symmetric protocol that enables one of several processors to eventually guarantee that all (or, in a more general setting, enough) other processors have acted, using a shared register but no global clock. Dropping the symmetry requirement gives a well-known exercise often phrased in terms of prisoners entering, in an unknown sequence, a room equipped with a single binary switch, and using it to communicate. Kane and Kominers recently analysed a more general version of the latter with multiple parallel and indistinguishable rooms. We answer some open questions of Kane and Kominers regarding the minimum number of switch states needed for the prisoners to solve the problem. We also consider the symmetric “wakeup” version of this scenario, and establish exactly for which numbers of processors and registers a solution is possible.

The number of finite additive 2-bases is known to grow exponentially. While this fact has been established by Marzuola and Miller (2010) using complex analytic techniques embedded in the study of numerical sets, we provide a direct, short proof using elementary probabilistic arguments.

We study the equidistribution of mesh patterns of length 2. We show that the number of equidistribution equivalence classes lies between 105 and 108, and conjecture that it is exactly 105. As a consequence, we obtain an upper bound of 49 Wilf-classes, improving the previously known bound of 56 due to Hilmarsson et al., and reducing the problem to three remaining conjectural equivalences (with the actual number conjectured to be 46). Our approach combines bijective constructions, generating functions, recurrence relations, and structural symmetries. We establish several new equidistribution results, including four previously unknown distribution classes, connect numerous patterns to known distributions in the literature, and resolve seven open pattern-avoidance enumeration problems posed by Hilmarsson et al. This work provides a near-complete classification of mesh patterns of length 2 and unifies several previously isolated results within a coherent framework.

Let $n$, $k$ and $t$ be positive integers, and let $\mathcal{F}$ be a collection of $k$-subsets of $[n]=\{1,2,\dots,n\}$. The $t$-covering number $\tau_t(\mathcal{F})$ of $\mathcal{F}$ is defined as the minimum size of a set $T$ such that $|F\cap T|\geq t$ for all $F\in \mathcal{F}$. For positive integers $k_1$ and $k_2$, let $\mathcal{F}_i$ be a collection of $k_i$-subsets of $[n]$ for $i\in \{1,2\}$. The families $\mathcal{F}_1$ and $\mathcal{F}_2$ are said to be cross $t$-intersecting if $|F_1\cap F_2|\geq t$ for all $F_1\in\mathcal{F}_1$ and $F_2\in \mathcal{F}_2$. When $\mathcal{F}_1=\mathcal{F}_2$, $\mathcal{F}_1$ is called a $t$-intersecting family. In this paper, we first characterize the extremal structures of cross $t$-intersecting families $\mathcal{F}_1$ and $\mathcal{F}_2$ that maximize $|\mathcal{F}_1||\mathcal{F}_2|$ under the condition that $\tau_t(\mathcal{F}_1)\geq t+1$ and $\tau_t(\mathcal{F}_2)\geq t+1$. We then describe the maximal $t$-intersecting families with $t$-covering number $t+1$.

We give an elementary, Fourier-free proof of Roth's theorem. The proof follows Roth's original density-increment strategy, but replaces the usual Fourier-analytic step with a direct combinatorial argument involving averages over sub-progressions.

A universal cycle is a cyclic sequence in which each object of a combinatorial family appears exactly once as a contiguous window. While such cycles are well understood for many discrete structures and linear subspaces, the case of affine lines presents additional difficulties arising from parallelism. We prove that universal cycles exist for affine lines in $\mathrm{AG}(n,q)$ for all $n \ge 2$ and all prime powers $q$. Our construction embeds the problem into $\mathrm{PG}(n,q)$, using points at infinity to encode directions, and proceeds via a decomposition into pairwise and triple configurations combined with a recursive lifting and gluing argument. We further interpret the construction in the Grassmannian $G_q(2,n+1)$, where affine lines correspond to the outer shell of $2$-subspaces, thereby extending known constructions for Grassmannians. A Python implementation is provided as supplementary material.

We study a distributed consensus problem on a complete communication network of $n$ vertices, each holding one of two opinions. The vertices communicate in rounds, possibly in the presence of adversarial noise, and exchange information until they all agree on a single opinion. We consider a general class of protocols, where the vertices randomly sample neighbors and update their own opinion according to an update function $f$ depending on the sampled opinions. A prominent example is the $k$-maj protocol, where every vertex adopts the majority opinion of $k$ randomly sampled neighbors, breaking ties uniformly. We consider the runtime $R_n$ that is the number of rounds until all vertices agree on the same opinion, which we call the dominating opinion $D_n$. In our main result we describe the limiting distributions of these two key quantities for a large class of update functions $f$, for arbitrary initial configurations and under the presence of an adversary who may alter the opinions of up to $o(\sqrt{n})$ vertices in each round. We show that there are $f$-specific constants $\gamma, m > 0$ such that $R_n$ centers around $\mu_n = \frac{1}{2}\log_\gamma n + \log_m\ln n$, and we describe the asymptotic distribution of $R_n - \mu_n$. In particular, we show that it does not converge, and that it becomes asymptotically periodic both in the $\log n$ as well as the $\log\log n$ scale. Applied to $k$-maj, our results show, among other things, that $\gamma_{k\text{-maj}} = \binom{k-1}{\lfloor k/2 \rfloor}2^{1-k}k \sim ({2k}/{\pi})^{1/2}$.

In this short note we revisit the upper bound of the asymptotic least density of covering codes of radius $R$ in $[q]^n$ established by Krivelevich, Sudakov, and Vu. We show that by using a slightly different optimization in their core theorem we can obtain a constant factor improvement to their upper bound.

This paper studies limit profiles for the separation distance. A limit profile records the limiting shape of the distance to stationarity inside the cutoff window, at times of the form $t_n+cw_n$. We start with two famous card shuffles, a general setup for inverse riffle shuffles and random transpositions, and we determine their separation distance limit profiles. We then develop a spectral comparison technique and study continuity properties in the style of [Nes24; Nes25], adapted to separation distance. The comparison method is illustrated through random transpositions, as well as random walks on product groups and the hypercube.

We say that an instance of a constraint satisfaction problem (CSP) is non-redundant if the satisfaction of each clause cannot be implied by the satisfaction of the other clauses in the instance. The non-redundancy (NRD) of a CSP is the maximal number of clauses a non-redundant instance can have for a given number of variables. NRD is closely tied to the behavior of CSPs in various computational models including their sparsification, kernelization, and streaming complexity. A primary open question in the study of non-redundancy is the identification of which CSP predicates have near-linear NRD. Recent works by Carbonnel [CP 2022], Khanna, Putterman and Sudan [STOC 2025], Brakensiek and Guruswami [STOC 2025] and Brakensiek, Guruswami, Jansen, Lagerkvist, and Wahlström [2025] have introduced various forms of gadget reductions between CSPs to relate their non-redundancy. The primary contribution of this work is to recontextualize many of these gadget reductions in a framework which we call hypergraph projections. By studying a quantity we call the shrinking factor of these hypergraph projections, we can more precisely predict when a gadget reduction between predicates can yield a super-linear NRD lower bound, greatly improving on the analysis of previous works. To illustrate the power of our framework, we identify some concrete CSP predicates whose non-redundancy is at the cusp of our understanding and show how our methods give lower bounds that could not have been achieved with previous methods. We also demonstrate how these gadget reductions can be automatically deduced using SAT solvers, thereby opening up novel computational avenues for discovering further relationships between the non-redundancy of various CSPs.

Random walks on graphs can be slow. To speed them up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon>0$ that we can choose it. We show that in this case, at least for graphs of bounded degree, there is a way to steer the walk so that it visits every vertex in $n^{1+o(1)}$ steps with high probability. The key to this result is a way to decompose arbitrary graphs into small-diameter pieces.

We establish discorrelation estimates between the Piatetski-Shapiro prime set \[ \mathcal{P}_{\gamma} := \{p \text{ is prime and } p = \lfloor n^{1/\gamma} \rfloor \text{ for some } n \in \mathbb{N}\} \] and arbitrary nilsequences when $\gamma \in (0,1)$ is sufficiently close to $1$. This extends earlier works which treated linear or polynomial exponential phase functions. As an application, we establish an asymptotic formula for the number of solutions in $\mathcal{P}_{\gamma}$ to any “finite-complexity" system of linear equations, including for the number of $k$-term arithmetic progressions in $\mathcal{P}_{\gamma}$ up to a threshold $N$ for any given $k \geq 3$. Furthermore, we show that there exists an absolute constant $C>0$ such that if \[ 1 - 2^{-Ck} < \gamma < 1, \] then the Piatetski-Shapiro primes $\mathcal{P}_{\gamma}$ contain infinitely many non-trivial $k$-term arithmetic progressions. This significantly improves upon the previous range of $\gamma$ obtained by Li and Pan, which is of triple exponential type.

We develop a homotopical framework for small categories that extends classical invarints of algebraic topology to the categorical setting. Our approach is based on the construction of genuine path category, obtained trough a localization procedure, which allows us to define strong and weak fibrations for functor. We establish their basic properties, introduce a fibrant replacement for functors, and extend homotopical invariants such as the Svarc genus and sectional category to small categories. Finally, we apply this framework to motion planning in small categories, providing categorical analogues of Farber's topological complexity while removing finiteness constraints typical of existing approaches.

A proper coloring $c$ of a simple graph $G$ is harmonious if, for every pair of distinct edges $uv,xy\in E(G)$, we have that $\{c(u),c(v)\}\neq \{c(x),c(y)\}$. The harmonious chromatic number of $G$, denoted by $h(G)$, is the least positive integer $k$ such that $G$ has a harmonious coloring with $k$ colors. In this work, we extend an idea presented in [Kolay, et al. Harmonious coloring: Parameterized algorithms and upper bounds. Theor. Comp. Sci. 772 (2019), 132-142] to compare the harmonious chromatic numbers of two graphs $G$ and $H$, with $H$ being obtained from $G$ by identifying vertices at distance at least three. Furthermore, by fixing a proof presented in the same work, we manage to improve one of its upper bounds. We also introduce and study the first, to the best of our knowledge, integer-linear programming formulations for this problem in the literature, along with some heuristics. We provide some preliminary tests on random instances and instances from the second DIMACS Implementation Challenge.

We prove that Sharma-Mittal entropy is a subadditive and supermodular function on the lattice of all $n$-dimensional probability distributions, ordered according to the partial order relation defined by majorization among vectors. Our result unifies and extends analogous results presented in the literature for the Shannon entropy, the Tsallis entropy, and the Rényi entropy.

We introduce and study integral planes associated with crystallographic and non-crystallographic integral systems in real composition algebras. For an integral order $\Order$ in such an algebra we define the plane $\Order^{2}$ with quadratic form $Q(x,y)=\NN(x)+\NN(y)$, the axis shell, the balanced shell, and the corresponding unit-normalised spherical polytopes. For ten crystallographic orders we recover, in one uniform construction, the orthogonal-direct-sum root systems $2A_{1}$, $A_{2}\oplus A_{2}$, $4A_{1}$, $D_{4}\oplus D_{4}$, $16A_{1}$, and $E_{8}\oplus E_{8}$ (with classical-polytope realisations including the square, the 16-cell, the 24-cell, and the Gosset polytope $4_{21}$); for two non-crystallographic orders we obtain $H_{2}\oplus H_{2}$ (decagonal tegum) and $H_{4}\oplus H_{4}$ (600-cell tegum) over $\Z[\golden]$. We prove a rank-obstruction theorem that closes, unconditionally and by a purely Coxeter-theoretic argument, the existence question for an indecomposable rank-eight golden octonion order: no such order can exist. On the balanced shell side, we identify the genuine algebraic Hopf map $\Hopfmap_{A}(a,b)=(2a\bar b,\NN(a)-\NN(b))$ and prove that its restriction to the balanced shell is a finite principal fibration of the unit loop, valid both for the associative case and for the alternative Moufang case.

Determining the number of realisations, up to isometries, of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that, asymptotically almost surely, the $d$-dimensional complex realisation number for each $n$-vertex graph in an Erdős-Rényi random graph process is either infinite or equal to $2^{n-t}$ where $t$ is the size of the $(d+1)$-core; moreover this number coincides exactly with the real realisation number for such graphs. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ positive semi-definite matrix completion problem with randomly selected non-diagonal unknown entries.

For integers \(r\ge 2\), \(t\ge 1\) and a real number \(a\in(3/2,2]\), we study the typical structure of oriented graphs and digraphs that do not contain a blow-up \(T_{r+1}^t\) of a transitive tournament. We prove that almost every \(T_{r+1}^t\)-free oriented graph on n vertices admits an r-partition \(V_1\cup\cdots\cup V_r\) such that each induced subgraph \(G[V_i]\) is \(T_2^t\)-free, and the same holds for almost every \(T_{r+1}^t\)-free digraph.Consequently, the number \(f(n,T_{r+1}^t)\) of labelled \(T_{r+1}^t\)-free oriented graphs satisfies \(f(n,T_{r+1}^t)=|\mathcal{P}_{n,r,t}|(1+o(1))\), where \(\mathcal{P}_{n,r,t}\) is the family of oriented graphs admitting such an r-partition with each part \(T_2^t\)-free; an analogous statement holds for digraphs.When \(t=1\) this recovers the result of K"uhn, Osthus, Townsend and Zhao (2017) that almost all \(T_{r+1}\)-free oriented graphs (resp. digraphs) are r-partite, thereby confirming a generalised form of Cherlin's conjecture. Our proof combines the hypergraph container method, a weighted Erdős-Stone theorem, and a stability analysis for near-extremal \(T_{r+1}^t\)-free digraphs.

A finite group $G$ is called a Schur group if every Schur ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of the symmetric group $Sym(G)$ that contains all right translations of $G$. The list of all possible abelian Schur groups was obtained by Evdokimov, Kovács, and Ponomarenko in 2016. In two papers, we complete a classification of abelian Schur groups. In the present paper, we prove that several groups of nonpowerful order from the list are Schur groups. By that, we obtain a classification of abelian Schur groups.

We develop a framework for analyzing parameter symmetries in deep ReLU networks and obtain a complete characterization of the generic parameter fibers for three-layer bottleneck architectures. Our approach provides explicit semi-algebraic descriptions of these fibers and yields a polynomial time algorithm for deciding functional equivalence of two parameters. The symmetries include discrete and continuous transformations arising from layer composition, and depend on whether deeper layers hide or preserve geometric structure from preceding layers. Finally, we show that some of these symmetries induce local conservation laws along gradient flow, while others do not.

In this paper, we establish that the maximum edit distance of an $n$-vertex graph from the hereditary property of word-representable graphs is $n^2/8-o(n^2)$. In addition, we establish that the maximum edit distance of an $n$-vertex graph from the hereditary property of poset comparability graphs is $5n^2/32-o(n^2)$. In fact, we determine the edit distance function over all edge densities $p\in [0,1]$ for the property of word-representable graphs, for the property of $k$-word-representable graphs for each $k\geq 2$, and for the property comparability graphs. The latter has a peculiar structure that requires an infinite sequence of colored regularity graphs.

We prove that if $\Gamma$ is a finite connected vertex-transitive cubic graph, then either $|V\Gamma| \le 90$, or $\Gamma$ is a split Praeger–Xu graph, or there exist two vertices $\alpha$ and $\beta$ such that the identity is the only automorphism of $\Gamma$ fixing both $\alpha$ and $\beta$.

Higher dimensional permutations are tuples of d-1 permutations that can be identified with a point set in a d-dimensional grid. In N. Bonichon and P.-J. Morel, J. Integer Sequences 25 (2022), several conjectures regarding the enumeration of pattern avoiding d-permutations were stated. In this paper, we consider a mapping from d-permutations to $2^{d-1}-$ary trees that naturally generalizes the classical max-tree construction for permutations. We then show that, when restricted to d-permutations avoiding (21,12) and 231, this mapping yields a bijection with d-ary trees. This result resolves one of the conjectures of Bonichon and Morel.

We provide another approach to Friedland's result that the topological entropy $h$ of a symmetric nearest-neighbor subshift is computable. Instead of the previous algebraic technique, our approach is mostly combinatorial and involves only counts of locally admissible patterns $C_n$ of a cube $[1,n]^d$ in $\mathbb Z^d$. The main idea is a reflection-gluing construction: we flip admissible patterns and merge them along their boundaries. In addition to a short and elementary proof, another advantage is that our approach yields an explicit convergence rate in arbitrary dimensions, whereas obtaining such a rate is already complicated for $\mathbb Z^3$ in Friedland's approach. In particular, we show that for every $n\ge 1$, \[ \frac{1}{n^d}(\log C_{n+1} - q_d(n)\log|\Sigma|) \le h \le \frac{1}{n^d} \log C_n, \] where $\Sigma$ is the alphabet and \[ q_d(n)=(2^d-1)\sum_{k=0}^{d-1} \frac{\binom{d}{k}}{2^d-2^k}\, n^k. \]

We investigate the class of FHP theories, i.e. theories of structures in which all definable families of sets satisfy the Fractional Helly Property (and its variants) from combinatorics. FHP theories generalize NIP and form a new subclass of low NTP$_2$ theories. We give many new examples (including ultraproducts of finite fields and of the $p$-adics) and establish some results about forking and $f$-generics for amenable groups definable in FHP theories. We make several conjectures about finitary combinatorial properties of forking in NTP$_2$ theories and establish some partial results, as well as investigate related two-cardinal type counting functions addressing a question of Adler.

A morphism is a mapping that transforms words through letter-wise substitution, where each symbol is consistently replaced by a fixed word. In the field of combinatorics on words, one topic that has attracted considerable attention is the characterization of morphisms that preserve specific properties, such as overlap-freeness, square-freeness, lexicographic order, and primitivity. Continuing this direction, we initiate the study on occurrence-preserving morphisms, which address the following fundamental question: given a morphism $\phi$, two words $u$ and $v$, and $k \geq 1$, under what conditions does the number of occurrences of $u$ in $v$ equal the number of occurrences of $\phi^k(u)$ in $\phi^k(v)$? To answer this question, we introduce the notion of interference-free morphisms, examine their properties, develop an efficient algorithm for deciding interference-freeness, and uncover a connection to recognizable morphisms. We then present a precise characterization of occurrence-preserving morphisms in terms of interference-freeness. As applications of our characterization, we first show that there exists a bijection between the starting positions of the occurrences of $u$ in $v$ and those of $\phi^k(u)$ in $\phi^k(v)$. We then apply the characterization to the Fibonacci and Thue-Morse words to identify their minimal unique substrings~(MUSs). Finally, we exploit the connection between MUSs and net occurrences to simplify existing proofs on net occurrences in these words.

Kupavskii proved a codegree version of the Erdős–Ko–Rado theorem by showing that for an intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n \geq 2k +3d/(1-d/k)$, the minimum $d$-degree of $\mathcal{F}$ is at most $\binom{n-d-1}{k-d-1}$. Huang and Zhang improved the bound on $n$ to $n \geq 2k+2d-3$. In this short note, we prove that if $d = k-1$, then the bound on $n$ can be improved to $2k + \sqrt{2k} + O(1)$. In addition, we extend our method to show that the bound on $n$ can be improved to $2k + 7k^{2/3}+O(k^{1/3})$ when $d=k-2$.

In this paper, we study Thrall's problem for the higher Lie modules $L_\lambda$. Our main result provides a tableau-theoretic description of the Schur expansion of the character of $L_\lambda$ when $\lambda$ has two rows, thereby solving Thrall's problem in this case. This formula is expressed in terms of standard Young tableaux with major index congruence conditions and a spin-parity condition defined through bijections with Yamanouchi domino tableaux. We also obtain tableau formulas for hook shapes and partitions with distinct parts, and these results extend to all partitions in which each part greater than $2$ occurs at most twice.

In this article, we study the zero-divisor graph of the commutative ring with identity $R= \mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p,$ where $u^2 = 0, v^2 = 0, uv = vu$ and $p$ is an odd prime. We determine several graph-theoretic properties associated with the zero-divisor graph $\Gamma(R),$ including the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth. In addition, we compute certain topological indices of the graph $\Gamma(R).$ Furthermore, we find the eigenvalues, energy and spectral radius of the adjacency matrix, the Laplacian matrix and the Eccentricity matrix of the zero-divisor graph $(\Gamma(R).$