Papers

In the transversal achievement game on the $n\times n$ board, two players alternately claim cells, and the first to own a transversal—a set of $n$ cells of which no two share a row or column—wins. Ran{\dj}elović showed that the first player wins for every $n\ge4$, while the game is a draw for $n=2,3$. We give an independent proof that the first player wins for $n\ge4$ that additionally establishes a bound on the length of the win: the given strategy forces a win by ply $2n+3$, i.e.\ on the first player's $(n+2)$-nd move, for every $n\ge4$. The proof yields a strategy that is fully determined by a fixed rule on the current position and can thus be implemented directly. We isolate the use of the hypothesis $n\ge4$ to two steps in the analysis, explaining why the argument fails at $n=3$. An exhaustive computational search implementing the strategy verifies it against every legal defense for $n=4,5,6$, confirming both the strategy's validity and that the $2n+3$ bound is attained in these cases. The main theorem has also been formalized and machine-checked in Lean 4.

We prove the conjecture of Donnelly et al. that a certain class of Littlewood-Richardson tableaux are equinumerous with peakless Motzkin paths of length $n$. Furthermore, we construct an explicit bijection between this class of tableaux and peakless Motzkin paths of length $n$ for all $n\ge 1$.

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph in $\mathcal{F}$. A wide range of relaxations of graph isomorphism arise this way: isomorphism itself over the class of all graphs [Lovász, Acta Math. Hung. 1967], equivalence under the $k$-dimensional Weisfeiler-Leman algorithm over the graphs of treewidth $\leq k$ [Dvořák, J. Graph Theory 2010], and quantum isomorphism over planar graphs [Mančinska-Roberson, FOCS 2020]. Since the class $\mathcal{F}$ is typically infinite, it is not clear a priori whether homomorphism indistinguishability over $\mathcal{F}$ is decidable; for planar graphs it is undecidable. Every class for which decidability was previously known is sparse. We give the first decidability results for dense graph classes: We introduce the dense Weisfeiler-Leman algorithm that decides homomorphism indistinguishability over the class of graphs of cliquewidth $\leq k$, the dense counterpart of treewidth. This relation was not previously known to be decidable. The algorithm colors $k$-tuples of vertex subsets rather than $k$-tuples of vertices. Beyond the class of all graphs of cliquewidth $\leq k$, we prove a general meta-theorem: homomorphism indistinguishability over every $\mathsf{CMSO}_1$-definable graph class of bounded cliquewidth is decidable, in randomized exponential time. For classes of bounded linear cliquewidth the bound improves to $\mathsf{PSPACE}$, and we show this is tight by exhibiting such a class for which the problem is $\mathsf{PSPACE}$-complete. These are the first general algorithms for homomorphism indistinguishability over dense graph classes.

The ring $R_m$ of $m$-symmetric functions consists of the formal power series that are symmetric in the variables $x_{m+1},x_{m+2},\dots$ but carry no symmetry in the first $m$ variables. We develop a combinatorial theory for the specialization at $t=0$ of the Schur functions of $R_m$. Our main tool is a new characterization of right key tableaux as suprema of the sets of decreasing subwords of the reading words of the subtableaux of $T$. Being invariant under elementary Knuth transformations, this characterization is compatible with the RSK correspondence. We obtain in this way a generating function over semistandard tableaux for the $m$-symmetric Schur functions at $t=0$, together with a combinatorial proof of a Cauchy identity in $R_m$. The $m$-symmetric Schur functions and their dual are then respectively identified with Demazure atoms and Demazure characters. Restricted to the last $m$ variables, our correspondence specializes to a proof, by ordinary RSK, of Lascoux's nonsymmetric Cauchy identity for Demazure characters and atoms. As further applications, we relate the $m$-symmetric Schur functions at $t=0$ to the almost symmetric Schur functions through a unitriangular change-of-basis matrix, obtain tableau generating functions and Cauchy identities for both families, and derive Jacobi-Trudi type determinantal formulas for three different bases.

This study applies finite automata to the automatic solving of a variety of combinatorial games. For games whose positions and moves can be represented as regular languages and their operations, we design a two-stage automatic solving algorithm: first, construct a candidate finite automaton to determine the $\mathcal{P}$- and $\mathcal{N}$-positions, and then perform rigorous formal verification on this automaton; once verified, a complete solution of the game is obtained. For partizan octal games, we introduce a generalized misère quotient, overcoming the limitation that traditional theory applies only to impartial games. Using the above algorithm, we successfully solve the majority of two-digit partizan octal games, and based on these results, we propose a partizan version of Guy's conjecture. We also successfully solve a considerable number of partizan octal games under misère play, and give a conjecture on the structure of those games exhibiting “algebraic periodicity” among them. For Kotzig's nim, we resolve the most important related conjecture: we prove that the outcomes and SG values are periodic under both normal and misère play (including their partizan versions). Our algorithm successfully solves several small-scale cases, including misère play and partizan versions. This study pioneers a new theoretical tool and algorithmic paradigm for the automatic solving of combinatorial games, and has broad prospects for further extension and application in the field of combinatorial game theory.

We consider Goldbach type sums corresponding to the Liouville function and prove the existence of sufficient cancellations. We also consider applications to sign patterns in the Liouville function.

For an integer $n\ge0$, let $f(n)$ be the minimum number of subcubes of $\mathbb{Z}_3^n$ of the form $A_1\times\cdots\times A_n$, where $|A_i|=2$ for every $i$, whose union covers $\mathbb{Z}_3^n$. A simple counting argument gives $f(n)\ge(3/2)^n$, while $f(n)=O(n(3/2)^n)$ by random construction. We prove that $f(n)\le2(3/2)^n-1$, answering a problem of Imre Leader. We also show that $f(n)/(3/2)^n$ is nondecreasing and there exists a constant $C_3$ such that $f(n)=(C_3+o(1))(3/2)^n$ where $1.62227<C_3\le2$.

Chapoton introduced an interesting family of polytopes called arbor polytopes, where a polytope $\mathcal{Q}_\tau$ is associated to any arbor $\tau$. Chapoton conjectured that the polynomial $h(\tau)$ which counts lattice points in $\mathcal{Q}_\tau$ by number of nonzero entries is palindromic and unimodal. Athanasiadis, Xiao, and Yan recently proved that $h(\tau)$ is gamma-positive for a certain class of arbors they call octopuses. Since their proof was computational, they asked for one that is bijective. We present such a proof. We also extend their results by showing that $h(\tau)$ is gamma-positive for a larger class of arbors we call lopsided octopuses.

The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway–Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.

In this paper, we first extend the Christoffel-Darboux formula for orthogonal polynomials to general three-term recurrence sequences, and then investigate the identities and congruences for $g_n(x)$ and $v_n(x)$ given by \begin{align*} &g_0(x)=1,\ g_1(x)=\frac{x+1}2,\ (n+1)^2g_{n+1}(x)=\Big(2n(n+1)+\frac{x+1}2\Big)g_n(x)-n^2g_{n-1}(x)\ (n\ge 1), \&v_0(x)=1,\ v_1(x)=x,\ (n+1)^3v_{n+1}(x)=(2n+1)(n(n+1)+x)v_n(x)-n^3v_{n-1}(x)\ (n\ge 1).\end{align*}

A set in $\mathbb{R}^d$ is $s$-convex if it is the union of at most $s$ convex sets. A family $F$ satisfies the $(p,q)$ property if among any $p$ sets in $F$, some $q$ intersect. Let $\mathrm{HD}_d^{(s)}(p,q)$ be the minimum number of points needed to pierce a finite family of $s$-convex sets that satisfies the $(p,q)$-property. Alon and Kalai (1995) proved that $\mathrm{HD}_d^{(s)}(p,q)$ exists for any $p \geq q \geq d+1$ and any $s \geq 1$, but the quantitative bounds they obtained are very loose. We present several improved upper and lower bounds, for a general $d$ and for $s$-intervals of the line (i.e., $\mathrm{HD}_1^{(s)}(p,q)$). In particular, we prove the following: (i) For every $d\ge2$, $s \geq 1$ and $\delta>0$, if $p>q$ and $q\ge C_d\log(e sp)$, then $\mathrm{HD}_d^{(s)}(p,q) \le p-q+1 + O_{d,\delta}((s \cdot \tfrac{p}{q} \cdot \log \tfrac{esp}{q})^{\rho_d+\delta}),$ where $\rho_d<d$ is the exponent in the weak epsilon-net theorem of Rubin (2022). (ii) For $s \geq 1$, $p \geq q \geq 2$ and $q\ge C_0s\log(2s)\log(ep)$, $p-q+s \leq \mathrm{HD}_1^{(s)}(p,q) \leq p-q+2s+1$. This result provides the first near-tight estimate for $\mathrm{HD}_1^{(s)}(p,q)$ for $q>2$. (iii) For any fixed $s$, there are an integer $\kappa_s\in\{s,\ldots,2s\}$ and constants $C_s,p_s>0$ such that, whenever $p\ge p_s$ and $q\ge C_s\log(ep)$, $ \mathrm{HD}_1^{(s)}(p,q)\in\{p-q+\kappa_s,\;p-q+\kappa_s+1\}. $ Interestingly, this two-value concentration result holds, although the exact value of the threshold remains unknown. (iv) For any $s \geq 1$, $\mathrm{HD}_3^{(s)}(p,4) \geq sp^{2-o(1)}$. Already for families of convex sets, this significantly improves the best known lower bound on $\mathrm{HD}_d^{(1)}(p,d+1)$, for all $d \geq 3$.

For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1})=2m\left(\frac{n}{m}\right)^m+O_m\!\left(n^{m-1/5}\right).$$ The proof uses a sharp weighted cycle–path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.

We consider the problem of shifting two tokens placed on nonadjacent vertices $u,v$ of a graph $G$ on $n$ vertices to two nonadjacent vertices $u',v'$ of $G$ using a sequence of token movements. In each step, a token is moved from the vertex it is on to a neighbour of that vertex, ensuring that the tokens remain on nonadjacent vertices after this move. We answer a question of Briański, Felsner, Hodor, and Micek [“Reconfiguring Independent Sets on Interval Graphs”, MFCS 2021] by showing that if the two tokens can be moved from their initial position to their final position, then it can be done using at most $4n$ moves.

Physical neural networks perform learning through the intrinsic nonlinear dynamics of matter. Optimising their design presents a considerable challenge: complex many-body physics can provide powerful computation, but are expensive to simulate and large experimental optimisation runs are impractical to fabricate. Hence, the high-dimensional space of possible network topologies cannot be effectively directly searched. Here, we show that this search can be efficiently performed in an abstract graph space that is vastly cheaper to explore. Using random lasing networks – composed of interconnected nanoscale waveguides and hosting strongly coupled lasing modes – as an exemplar physical vision system, we establish a quantitative three-layer link: simple graph-theoretic metrics predict the nonlinear lasing physics, which in turn predicts vision performance. After validating this relationship using physical simulations, we exploit it to drive an evolutionary algorithm using graph metrics, producing network topologies that outperform random designs at a fraction of the computational cost (3000$\times$ speed-up compared to physical simulation). On simulated image-classification tasks, graph-optimised networks substantially improve classification accuracy. As our framework operates on network topology rather than substrate-specific physics, we anticipate it can transfer to other network-based physical learning systems, providing an efficient route for the directed design and optimisation of complex, strongly-interacting physical neural networks.

A Deza graph is a regular graph in which the number of common neighbours of two distinct vertices takes at most two values, regardless of adjacency. Its children are the graphs on the same vertex set in which adjacency is determined by these two common-neighbour counts. A Deza graph is called strongly Deza if both children are strongly regular. We construct an infinite family of edge-regular strongly Deza graphs using non-degenerate quadratic forms over the field with five elements. For every odd dimension greater than three and each of the two determinant square classes, we obtain a graph on the projective points represented by vectors of norm one. Its children are complementary strongly regular graphs with the parameters of the corresponding orthogonality graph on non-isotropic points and its complement. We determine the parameters by counting solutions to systems involving the associated bilinear form. These counts also yield symmetric association schemes over finite fields of odd characteristic. Further constructions include Deza graphs in dimension four, orthogonality graphs in odd dimensions, unions of relations in even dimensions over the field with nine elements, and an odd-dimensional family over the field with thirteen elements. We also give low-dimensional examples, including one whose children are a triangular graph and its complement.

We study a cooperative search game with $N$ lockers, $N-1$ labelled objects, one empty locker, and $N-1$ players. Player $i$ seeks object $i$ and may inspect at most two lockers; the second inspection may depend on the content of the first. The players may coordinate beforehand but receive no information about the searches of other players after play begins. We prove that the maximum probability that every player finds the assigned object is $I_N/N!$, where $I_N$ is the number of involutions of $N$ elements. An optimal strategy represents the blank by the common fictitious symbol $N$ and follows pointers: player $i$ first opens locker $i$, then opens the locker whose number was observed. The upper bound holds for every deterministic or randomized adaptive strategy and follows from a deletion lemma and two recurrences. We give explicit examples, identify the three-door case with an isomorphic "Return of Monty Hall" game, and report a reproducible mixed-integer verification for $N=2,3,4,5$. The computation is independent of, and not needed for, the proof.

For an $n$-vertex graph $G$ and a permutation $\pi$ of its vertex set, let \[ I_G(\pi)=|E(G)\cap E(G_{\pi})|,\qquad \mu(G)=\min_{\pi} I_G(\pi), \] where $G_{\pi}$ is the copy of $G$ obtained by relabelling every vertex $x\in V(G)$ as $\pi(x)$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph $G$ satisfying $\mu(G)\ge k$. Erdős recorded a construction of Mullin showing $f(n,5)\le 2n-2$ and asked whether equality holds for sufficiently large $n$. We prove that it does: \[ f(n,5)=2n-2 \] for all sufficiently large $n$. The proof strategy is a core–buffer–completion framework: it moves the few high-degree vertices into carefully chosen low-degree positions, confines the allowed overlap to this bounded part, and then relabels the sparse remainder without creating any additional common edge.

The Foregger-Sinkhorn tie-point conjecture asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero is larger than its permanent, then that zero is a tie point. We give a counterexample in dimension eight.

The restricted edge-connectivity of a graph is the minimum size of an edge-cut whose removal leaves every component with at least two vertices. In 2024, Zhang et al. showed that the restricted edge-connectivity of any optimal $1$-planar graph belongs to $\{8,10,12\}$. In this paper, we exclude $8$ as a possible value, thereby proving that the restricted edge-connectivity is either $10$ or $12$, and both values are attainable. Furthermore, we show that the restricted edge-connectivity of a 6-connected optimal 1-planar graph equals $10$ if and only if the graph contains an edge whose two endvertices both have degree $6$. As a key ingredient, we characterize the structure of vertex-induced subgraphs on $n$ vertices with $4n-9$ edges in optimal 1-planar graphs, and use this characterization to establish a connection between restricted edge-cuts and vertex-cuts in optimal 1-planar graphs.

A full-homomorphism from a graph $G$ to a graph $H$ is a function on vertex sets that preserves adjacency and non-adjacency of vertices. A graph $G$ is called a minimal $H$-obstruction if it has no full-homomorphism to $H$ but every proper vertex induced subgraph of $G$ does. Such graphs can have at most $|V(H)|+1$ vertices. The set of minimal $H$-obstructions on $|V(H)|+1$ vertices is denoted by $\mathrm{obs*}(H)$. In question 2 of the paper "Santiago Guzmán-Pro, Full-homomorphisms to paths and cycles, Discrete Mathematics, 347(3):113800, 2024" it is asked if there is a characterization of those graphs $G$ that lie in $\mathrm{obs*}(H)$ for some graph $H$. In this paper, we give a complete answer to this question.

The well-known Bermond-Thomassen conjecture states that every digraph of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint directed cycles. Despite being posed in 1981, this conjecture remains unresolved for all $k \ge 4$. We prove a relaxation of this conjecture: every digraph $D$ of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint cycles, each of which either is directed or can be made directed by reversing one of its arcs. This bound is sharp and answers a question raised by Cames van Batenburg during the online workshop "Entropy Compression and Related Methods" in $2021$.

Identifying and constructing graphs that are determined by their generalized spectrum (DGS) is a significant and challenging problem in spectral graph theory. Recently, a simple criterion for almost controllable graphs to be DGS was proposed by Lin et al. (2026), utilizing the modified walk matrix. In this paper, we investigate the evolution of the modified walk matrix under disjoint union and join operations with a singleton vertex. We establish an exact algebraic identity for the determinant of the modified walk matrix of the resulting graph. Based on this identity and the DGS-criterion of Lin et al., we construct infinite families of almost controllable graphs that are DGS, extending the previous construction of Liu et al. (2019), which was restricted to controllable graphs.

Let \(X\) and \(Y\) be finite sets with \(|X|=n\), \(|Y|=m\), equipped with sigma algebras \(\mathcal A\) and \(\mathcal B\). For arbitrary sigma algebras \(\mathcal A\) on \(X\) and \(\mathcal B\) on \(Y\), we enumerate measurable functions \(f\colon X\to Y\). When \(\mathcal B\) is discrete, the number of pairs \((\mathcal A,f)\) is the Touchard polynomial \(T_n(m)=\sum_k S(n,k)m^k\). For general \(\mathcal B\) with atom sizes \(b_1,\dots,b_r\), the number of pairs \((\mathcal A,f)\) over all sigma algebras \(\mathcal A\) on \(X\) is the complete Bell polynomial \(N_{\mathcal B}(n)\) in the power sums \(p_a=\sum_j b_j^a\), with exponential generating function \(\exp(\sum_j(e^{b_jx}-1))\). This specialises to the Touchard polynomial in the discrete case and is maximised by the trivial codomain sigma algebra. We further show that \(N_{\mathcal B}(n)=\mathbb E[Z^n]\) for a compound Poisson random variable \(Z\), and we discuss basic asymptotic growth of \(N_{\mathcal B}(n)\).

A vertex of a graph is called a top vertex if it has maximum degree in the graph. A vertex of a graph is called a Gallai vertex if it belongs to every longest path of the graph. Golan and Shan proved that every top vertex of any connected induced-$2P_2$ free graph is a Gallai vertex. Long, Milans, and Munaro subsequently showed that if every connected induced-$H$ free graph has a Gallai vertex, then $H$ must be a linear forest of order at most nine. They also proved that, for every linear forest $H$ of order at most four, every top vertex of any connected induced-$H$ free graph is a Gallai vertex. In this paper, we determine the graphs $H$ for which every top vertex of any connected induced-$H$ free graph is a Gallai vertex. Our result shows that this holds precisely when $H$ is a linear forest of order at most four or $H=P_3+2P_1$.

A brick is a 3-connected graph $G$ such that $G-u-v$ has a perfect matching for any two distinct vertices $u,v\in V(G)$. An edge $e$ in a matching covered graph $G$ is removable if $G-e$ is matching covered. We say that a removable edge $e$ in a brick $G$ is $b$-invariant if $b(G-e)=b(G)=1$, where $b(H)$ denotes the number of bricks in the tight cut decomposition of a matching covered graph $H$. An edge of a graph is solitary if it lies in precisely one perfect matching. Lucchesi and Murty proposed the problem of characterizing bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. Note that every $b$-invariant edge is removable. In this paper, we strengthen the condition by requiring that every removable edge is solitary. We show that every nonsolid brick satisfying this strengthened condition can be obtained by repeatedly splicing odd wheels (up to multiple edges). Moreover, properties of such bricks imply that "repeatedly splicing odd wheels" cannot be replaced by "repeatedly splicing copies of $K_4$".

For a connected graph $G = (V, E)$, a bijective edge labeling $f:E \to\{1,\ldots ,|E|\}$ is a local antimagic labeling of $G$ if it induces a vertex labeling $f^+$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(xu)$, with $u$ ranging over all the vertices adjacent to $x$. The minimum number of distinct induced vertex labels over all local antimagic labelings of $G$ is the local antimagic chromatic number of $G$, denoted $\chi_{la}(G)$. In this paper, we make use of algebraic analysis to obtain necessary and sufficient conditions for every bipartite graph with all vertices of degree 2 except exactly three vertices of degree at least $3$ (say, $u,v,w$, such that every $(u,v)$-, $(u,w)$- and $(v,w)$-path is of even length and not all are of length 2) to have local antimagic chromatic number 2. Moreover, we showed that the consecutive edge labels of every induced path of each case is unique.

Given a family of graphs $\mathcal{F}$, we define a graph $G$ to be fully $\mathcal{F}$-colourable if $G$ admits a full homomorphism to some $F$ in $\mathcal{F}$. We approach the problem of determining when a graph is fully $\mathcal{F}$-colourable in terms of minimal forbidden induced subgraphs. We provide general results which allow to obtain the exact families of forbidden induced subgraphs for full $\mathcal{F}$-colouring when $\mathcal{F}$ is among some well-known families, such as threshold, trivially perfect, split, chordal, interval and strongly chordal graphs, as well as forests. Traditionally, these questions have been studied for a single graph $H$, not a family. Motivated by our results on the family of forests, we contribute to this research by focusing on the case of a single centipede.

Let $f(t)\in\mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv1\pmod{k}$, we study cyclic orderings of subsets $A\subseteq \mathbb{F}_p$ for which $ f(a_i+a_{i+1})$ is a nonzero $k$-th power for every consecutive pair. By combining mixed character-sum estimates, Fourier analysis on $\mathbb{F}_p$, and spectral graph methods, we establish a threshold $c(p,k,f)$ such that every subset $A$ with $\#A\ge c(p,k,f)$ admits such a cyclic ordering. We also give lower and upper bounds for the optimal threshold.

We give a convex-geometric formula for the multiplicity sequence of a monomial ideal in terms of mixed volumes of polytopes constructed from its Newton polyhedron. We also construct a counterexample to a conjecture of Achilles and Manaresi proposing a different volume formula for the multiplicity sequence. Finally, we derive a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals.

Wood~ recently showed that every graph $G$ of pathwidth $h$ and $\Delta(G)\ge1$ admits a $T$-partition of width at most $4(h+1)^2\Delta(G)$ for some tree $T$ with $pw(T)\leq2h+1$. In this paper, we establish an analogous result for tree-depth, which is a stronger parameter than pathwidth. We prove that every connected graph with tree-depth $h$ admits a $T$-partition of width at most $\mathrm{max}\{1, (4h-10)\Delta(G)+1\}$ for some tree $T$ with $\operatorname{rad}(T)\leq h-1$.