Papers

For $n\ge5$, let $F\colon\mathbb{F}_2^n\to \mathbb{F}_2^n$ be almost perfect nonlinear and write $N=2^n$. It is proven that the linearity $\mathcal{L}(F)$ of $F$, i.e., the largest absolute Walsh coefficient of a nonzero component, is at most $N-10$ in even dimension and at most $N-6$ in odd dimension. This improves the general upper bound of $N-6$ in even dimension and $N-4$ in odd dimension. It is further proven that, for each fixed $k$, the $k$-th largest absolute Walsh coefficient among nonzero components, counted with multiplicity, is at most $(1+O_k(2^{-n}))N/\sqrt{k}$ as $n\to\infty$. The second and fourth largest coefficients are at most $2\lfloor N/3\rfloor$ and $N/2$, respectively. Finally, a bound on the linearity in terms of the number $q$ of nonplateaued nonzero components is derived. In odd dimension, for $q>0$, we have $\mathcal{L}(F)^2\le N(1+\sqrt{q(N-1)})$, so that $\mathcal{L}(F)/N\to1$ implies $q/N\to1$. In even dimension, for every $1/2\le C<1$, the condition $q\le(4C-C^2-1)N/4+1$ implies $\mathcal{L}(F)\le CN$. In particular, $q\le3N/16+2$ implies $\mathcal{L}(F)\le N/2$.

Imagine there are $n$ cars lined up along a one-way street containing $n$ spots. Each car contains a group of friends, including a reluctant driver. Each car has a preferred spot and cars enter one by one. The cars drive to their preferred spot and if it is empty park there. If it is not empty, a friend in the back yells out “Hey! You should have driven faster!". Frustrated by this, the driver continues down the road until they find the last unoccupied spot (if one exists) and parks there. We say a sequence $(a_1,a_2,\dots, a_n)$ of preferred spots is a frustrated parking function if all cars can park under this rule. In this paper, we study the enumerative properties of frustrated parking functions. In particular, we show that the number of frustrated parking functions of length $n$ is $(2n-1)!!$. This is done by associating frustrated parking functions with height labeled Dyck paths. Using this association, we are then able to better understand the sets of lucky cars and lucky spots for frustrated parking functions. We show that the frustrated parking functions of length $n$ where the first $k$ cars (or first $k$ spots) are lucky is given by $k!S(n,k)$ where $S(n,k)$ is the Stirling number of the second kind. This in turn implies that the number of frustrated parking functions where once a car (or spot) is unlucky, the remaining cars (or spots) are unlucky is counted by the $n^{th}$ Fubini number. We also show that the number of frustrated parking functions with $k$ lucky cars (or spots) is given by the second order Eulerian number.

We classify the maximum scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^5)$, proving that every such set is of pseudoregulus type or of Lunardon-Polverino type. Building on the reduction obtained by Lia, Longobardi and Zanella in [S. Lia, G. Longobardi and C. Zanella, Towards the classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$, Algebraic Combinatorics 9 (2026), 327-355], we show that the two remaining candidate families contain no scattered linear sets. Our approach reduces these candidates to two normal forms and establishes their non-scatteredness through the existence of rational points on associated algebraic varieties.

We prove that the $g$-polynomial of every split matroid has nonnegative coefficients, establishing Speyer's conjecture for a class closed under taking minors and containing all paving and copaving matroids. Our proof uses a deletion–contraction identity obtained by constructing an auxiliary split matroid. We first show that every simple, cosimple, connected split matroid $M$ has an element $e$ for which both the deletion $M\setminus e$ and the contraction $M/e$ are connected. More generally, let $M$ be any connected split matroid of rank $k$ on a ground set $E$ with $|E|\ge4$. Suppose that $e\in E$ is such that both $M\setminus e$ and $M/e$ are connected. For every $v\in E\setminus\{e\}$, we construct a connected elementary split matroid $N_{e,v}$ of rank $k-1$ on $E\setminus\{e,v\}$ satisfying \[ g_M(t)=g_{M\setminus e}(t)+g_{M/e}(t)+t\,g_{N_{e,v}}(t). \] Using a fixed total order on $E$, we prescribe the proper cyclic flats of $N_{e,v}$ and their ranks. The identity follows from the covaluative formula of Ferroni and Schröter together with recurrences for its correction polynomials, derived from Ferroni's enumeration of admissible Delannoy paths for Schubert matroids. Since all three matroids on the right have fewer elements, the identity supplies the induction step in the proof of nonnegativity.

Zero forcing is a process described by a color change rule on the vertices of a graph. In this paper, we maximize the discrepancy between various zero forcing parameters relative to graph order. First, we find an upper bound on the difference in cardinality between minimal zero forcing sets (sets containing no proper zero forcing subset) of maximum and minimum size, and we show that this bound is sharp for an infinite family of graphs. Furthermore, we derive an upper bound for the discrepancy between the maximum and minimum propagation times of the minimum zero forcing sets of any graph, showing this bound is sharp for an infinite family of graphs.

We study classical and quantum indistinguishability of sums of independent random permutations and related transformations from permutations to functions. Let $G$ be a finite abelian group of order $N$, and let $\pi^k_+(x)=\pi_1(x)+\cdots+\pi_k(x)$ for $k\geq2$ independent uniform random permutations of $G$. We give a unified Fourier analytic treatment in which the construction is represented by its probability density and a distinguisher by its acceptance function, with the classical and quantum query models imposing different restrictions on the Fourier support of the latter. Classically, we obtain the bound $O_k(q/N^{k-1/2})$ for every $q<N$, and refine it below the birthday threshold to $O_k(q^2/N^k)$. In the quantum model, a simulation argument gives $O_k(N^{-(k-3/2)})$ for $q\leq(N-1)/2$, while Fourier interpolation gives concrete finite bounds up to $q\leq4N/15$ and the query-dependent bounds $O\left(\min\left\{N^{-1/2},q^3/N^2 + 1/N\right\}\right)$ and $O_k\left(\min\left\{q^3/N^k,N^{-(k-3/2)}\right\}\right)$, for $k=2$ and $k \geq 3$, respectively, throughout $1\leq q\leq(N-1)/2$. For $q = 1$, the first bound sharpens to $O(N^{-2})$. Over $G=\mathbb F_2^n$, a one-query Fourier attack matches the order of our one-query bound, while an $N/2$-query parity attack with advantage $1/2$ shows that our bounds reach the constant-advantage query threshold. We further study two variants of sum of permutations over binary vector spaces. First, we allow arbitrary surjective linear postprocessing, which includes truncation, and obtain classical and quantum bounds that retain the output-size dependence. Second, we analyse Dinur's variable-output single-permutation construction, $\mathsf{LXoP}$, for every fixed output width, and derive its classical and quantum security bounds; for one- and two-block outputs, we give concrete quantum security bounds.

Let $\lambda_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph $G$ with positive degrees, where $k$ is an integer satisfying $1\le k\le |V(G)|$. Let $\varphi_k(G)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove $\varphi_k(G)\le C\sqrt{\lambda_k(G)\log(k+1)}$ for an absolute constant $C$. The number of sets and the spectral index are both $k$, and conductance is measured in the original graph. The logarithmic dependence is optimal up to an absolute constant. The proof combines geometric partitioning of the spectral embedding with minimum-cut improvement and adaptive projections in coefficient space. A single conductance threshold is used throughout the construction. The resulting maps have disjoint supports, and the sum of their Gram matrices is bounded below by an absolute positive multiple of the identity. A dyadic maximal estimate bounds the sum of their internal energies uniformly over unit coefficient vectors. A dimension argument using local eigenvalues then yields exactly $k$ disjoint sparse cuts.

Hadwiger's conjecture (1943) states that every graph $G$ with chromatic number at least $t$ contains a $K_t$-model: a collection of $t$ vertex-disjoint connected subgraphs $T_1,\dots,T_t$ such that for all $1\le i<j\le t$ some vertex in $T_j$ has a neighbour in $T_i$. Replacing "some" in this definition by "every" gives rise to the significantly stronger notion of a dominating $K_t$-model introduced by Illingworth and Wood (2024). They raised the question whether every graph of chromatic number at least $t$ contains a dominating $K_t$-model. This statement is a significant strengthening of Hadwiger's conjecture and has come to be known as the dominating Hadwiger conjecture. We provide our own exposition of a disproof of this conjecture found by ChatGPT 6 Astra Ultra. The construction is the complement of a pseudorandom triangle-free graph that is obtained by randomly subsampling a block geometric graph based on the Suzuki-Tits ovoid.

An antimagic labelling of a graph G = (V,E) is a bijection from E to {1,2,...,|E|}, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A graph is antimagic if it has an antimagic labelling. Hartsfield and Ringel in 1990 stated the celebrated conjecture: Every connected graph other than K_2 is antimagic. We prove the conjecture for join graphs with at least three vertices.

For vertex-transitive Ramanujan graphs with logarithmic girth, a simple random walk bridge of length of order $\log N$, where $N$ is the size of the graph, has a maximum distance that changes from order $\sqrt{\log N}$ to order $\log N$ in a bounded critical window. We prove this by separating bridges whose lifts to the regular tree close from those whose lifts do not. A uniform two-term return estimate determines the probabilities of these two cases and the real-valued critical center. In the same $O(1)$ window, the normalized range has a two-point limiting law whose mixture weights vary nontrivially across the window.

For a finite group $G$, let $\lambda(G)$ denote the minimum number of orbits on the elements of a finite lattice $L$ with $\operatorname{Aut}(L)\cong G$. Babai and Goodman conjectured that $\lambda(G)$ is bounded by an absolute constant. We prove that $\lambda(G)\leq 50$ for every finite group $G$, thereby confirming their conjecture. Moreover, the lattice can be chosen to have a regular orbit. The main algebraic ingredient is a decomposition of a generating set of an arbitrary finite $2$-group into an elementary abelian part and two sets in which no quotient of distinct elements is an involution.

We prove a sharp Szemerédi-Trotter estimate \[\mathcal{I}(A,\mathcal{L})\lesssim \frac{|A||\mathcal{L}|}{p}+|A|^{2/3}|\mathcal{L}|^{2/3}+|A|+|\mathcal{L}|\] for prime finite field $\mathbb{F}=\mathbb{F}_p$ by a new polynomial decomposition theorem. As applications, we first prove the sharp Furstenberg set estimate in $\mathbb{F}^2$. Secondly, we improve sum-product estimate \[\max\{|A+A|,|A\cdot A|\}\gtrsim\min\{(p|A|)^{1/2},|A|^{5/4}\},\quad A\subset\mathbb{F}.\] Finally, we improve the Fourier restriction estimate $R^*(2\to\alpha)$ holds for $\alpha>\frac{10}{3}$ in $\mathbb{F}^3$ when $p\equiv 3\mod 4$.

Bourneuf, Charbit and Thomassé [BCT25] showed that the domination number of a tournament can be bounded as a function of its fractional dichromatic number. The function proved was exponential and the tools were based on VC-dimension. In this paper, we present two new proofs of this theorem. The first proof is based on a reduction to the problem of bounding the domination number of a $(1/2-\epsilon)$-majority tournament, for which [BCT25] and Charikar, Ramakrishnan and Wang [CRW26] gave tight bounds. This proof yields the same exponential bound on the domination number as in [BCT25]. The second proof gives a quasilinear bound for the domination in terms of the fractional dichromatic number. It was obtained via AI and was inspired by the recent book proof of the existence of a Condorcet Winning Set of size five due to Ramakrishnan [Ram26].

We study the hard-core model on recursively defined sequences $(G_n)_{n\geq0}$ of graphs with a fixed number $k\geq 1$ of labeled vertices in each graph. The next graph in the sequence is constructed by taking a fixed number $m\geq 2$ of copies of the previous graph, connecting these copies by identifying some labeled vertices according to a fixed rule, and afterward choosing $k$ labeled vertices in the resulting graph, again in accordance with a fixed rule. Examples of such sequences include the Sierpiński gasket graphs, hierarchical lattices, and many more. We prove that, when the vertex degrees of the graphs $G_n$ are uniformly bounded and the distances between the labeled vertices in $G_n$ diverge, the complex zeros of the univariate independence polynomials $Z_{G_n}(\lambda)$ avoid a neighborhood of the non-negative real axis. By the Lee–Yang theory this implies that no phase transitions occur for the hard-core model on these recursive sequences of graphs, independently of the starting graph $G_0$. The proof relies on the study of the dynamical properties of a one-parameter family of rational maps $F_\lambda$ on the $(2^k-1)$-dimensional complex projective space induced by the graph recursion operator. The dynamical framework developed in this paper can be naturally extended to other classical models in statistical mechanics (such as the Ising or Potts models) and to more general notions of graph recursions.

The set-coloring Ramsey number $R(k;r,s)$ is the least $N$ such that every assignment of an $s$-element subset of $[r]$ to each edge of $K_N$ yields a copy of $K_k$ whose edges share a common color. For every fixed prime power $q$, we construct infinitely many positive integer triples $(r,j,s)$ with $j\sim(q-1)^{-2/3}r^{1/3}$ and $s=(1-1/q)(r-j)$ such that $R(q+1;r,s)=\Theta_q(r^{4/3})$. For $q=3$, this answers in the affirmative a question of Conlon, Fox, Pham and Zhao, showing that polynomially superlinear growth for $R(4;r,2(r-j)/3)$ already occurs at the scale \(j=\Theta(r^{1/3})\). Moreover, along the same sequence, the maximum size of a $q$-ary code of length $r$ and minimum Hamming distance at least $s$ is $(1+o(1))(q-1)^{4/3}r^{4/3}$.

Low-power error-correcting cooling (LPECC) codes and constant-power error-correcting cooling (CPECC) codes provide error correction while controlling power consumption and thermal effects in on-chip buses. In this paper, we study binary CPECC and LPECC codes with \(e=w-3\). For CPECC codes, we extend the applicability of the upper bound previously obtained by Zhao and Zhang from the quadratic-order condition \(w\ge 2t(t+1)+2\) to \(w\ge w_0(t)\), where \(w_0(t)\sim \sqrt{2}\,t^{3/2}\). Using Steiner systems, we show that the CPECC bound is attainable and asymptotically tight for fixed \(t,w\). For LPECC codes, we establish the new upper bound \(\left\lfloor\frac{\binom{n+2}{3}}{\binom{w+t}{3}}\right\rfloor\) for \(w\ge \mu(t)\), where \(\mu(t)\sim \sqrt{2}\,t^{3/2}\). This bound is strictly smaller than the previous bound of Zhao and Zhang whenever both apply, and is asymptotically tight for fixed \(t,w\) in the stated range.

The generalized packing–covering conjecture of Elimelech, Firer and Schwartz asserts that, for every linear code $\mathcal{C}$ and every admissible order $t$, the $t$-th generalized Hamming weight $d_t(\mathcal{C})$ and the $t$-th generalized covering radius $R_t(\mathcal{C})$ satisfy $d_t(\mathcal{C})\le 2R_t(\mathcal{C})+2$. We give a computer-assisted proof of the conjecture for every linear code over every finite field and every admissible order. Combining a parity-check reformulation of the conjecture, bounds on the length of putative counterexamples, and successive puncturing arguments, we settle all orders $t\ge 32$ and reduce the remaining orders to finitely many parameter tuples, which we exclude by an exact computer verification.

Let $F$ be a fixed graph without isolated vertices. An edge-colored graph is $F$-rainbow saturated if it contains no rainbow copy of $F$, but the addition of any missing edge in any color creates a rainbow copy of $F$. The rainbow saturation number $rsat(n,F)$ is the minimum number of edges in such a graph on $n$ vertices. We prove a dichotomy governed by isolated edges: if $F$ contains an isolated edge, then $rsat(n,F)=O(1)$ for all sufficiently large $n$, while if $F$ has no isolated edge, then $rsat(n,F)=\Theta(n)$. The linear lower bound is expressed in terms of a directed weight parameter $\eta(F)$ and establishes the linear half of the dichotomy; in several cases it also strengthens the Cameron–Puleo type coefficient. For the bounded half, we construct rainbow saturated graphs for targets of the form $H\cup K_2$. As an application of these constructions, we determine the asymptotically tight behavior for the rainbow saturation number of the generalized friendship graph $F_{t,p,q}=tK_p\vee K_q$, proving that $ rsat(n,F_{t,p,q})=(p+q-1)n+O(1)$ for fixed $t\geq 2$, $p\geq 2$ and $q\geq 1$ as $n\to\infty$.

Completely positive functions are an extension of completely positive matrices. They are known to characterize maximal spherical codes and maximum-density distance-avoiding subsets of $\mathbb{R}^n$ and certain compact metric spaces. This thesis expands this framework to related classes of problems in finite measure spaces and to the sphere-packing problem. For the latter, this is sharpened to show that the optimal sphere-packing density can be approximated using Schwartz functions. Converging hierarchies of semidefinite programming bounds on the size of optimal spherical codes are known, based on approximations of completely positive functions and the Lovász theta number of a graph. This thesis extends these hierarchies to distance-avoiding sets and similar problems and to the sphere-packing problem, and proves their convergence to the maximum density. For distance-avoiding sets, additional hierarchies, such as the moment hierarchy, are introduced and shown to be stronger than the completely positive hierarchy, hence they also converge. Related hierarchies for compact packing problems are also investigate. These bounds are implemented for Witsenhausen's problem, which asks for the maximum fraction $\alpha_n$ of the $n$-dimensional unit sphere that is coverable by a set avoiding orthogonal pairs; and for the $t$-almost-equiangular-set problem: finding the maximum size $\alpha(n,t)$ of a subset of the $n$-dimensional unit sphere in which every triple contains a pair with inner product $t \in [-1,1)$. An analytic solution to this bound yields an enumeration of optimal constructions for $n = 2$ and $3$ when $t \geq 0$.

A graph $G$ is point determining, if no two vertices have the same neighborhoods. In this paper, we show that this property is recognizable from the deck of cards of a graph.

We give a new proof that if the natural numbers are colored in two colors, then there exists a monochromatic Pythagorean triple. The proof was found by ChatGPT Astra.

For a graph \(F\), the Turán number \(\ex(n,F)\) is the maximum number of edges in an \(F\)-free graph on \(n\) vertices. Let \(q\ge 2\) be an integer and set \(n=q^{2}+q+1\). Brown and Erdős, Rényi and Sós independently proved that $\ex(n,C_{4})\ge \frac12 q(q+1)^{2}$ for every prime power \(q\), and Füredi subsequently established the upper bound $\frac12 q(q+1)^{2}$ for $\ex(n,C_{4})$ whenever \(q\notin\{1, 7,9,11,13\}\). In this article, we prove that every \(C_{4}\)-free graph \(G\) on \(n\) vertices with at most \(\frac12 q(q+1)^{2}\) edges satisfies $\tau(G)\le n^{(n-3)/2}$, where \(\tau(G)\) denotes the number of spanning trees of \(G\). In particular, for every prime power $q\notin\{7,9,11,13\}$, the above upper bound on $\tau(G)$ is attained precisely by the orthogonal polarity graphs, thereby proving London's conjecture for all such $q$.

Let $f(z)=\sum_{k=0}^{n}a_kz^k$ be a polynomial with positive coefficients, and define its coefficientwise reciprocal by $f^{\#}(z)=\sum_{k=0}^{n}\frac{z^k}{a_k}.$ It is known that if $f$ is palindromic and has only negative real zeros, then all zeros of $f^{\#}$ lie on the unit circle. We prove that if $p$ and $q$ are palindromic polynomials of degrees $n$ and $n+1$, respectively, with only negative real zeros, then their coefficientwise reciprocals have only simple zeros, and $p^{\#}$ strictly interlaces $q^{\#}$ on the unit circle. Our proof is based on finite Blaschke products and the comparison of their boundary phases. As an immediate consequence, we obtain strict interlacing for the reciprocal binomial, reciprocal Eulerian, and reciprocal Narayana polynomials. By combining a sign change in the $\gamma$-coefficients with coefficientwise reciprocation, we further construct strictly interlacing families from Rogers–Szegő, Poupard, and Kreweras-related polynomials.

Spencer's 1985 “six standard deviations suffice” theorem shows that every $A \in [-1,1]^{n \times n}$ has a sign vector $x \in \{-1,1\}^n$ with $\|Ax\|_\infty \le 6\sqrt{n}$. We show the upper bound $\sqrt{3\operatorname{arsinh}(10)}\sqrt{n}+4 < 2.9992 \sqrt{n} + 4$ by directly rounding the minimizer of a potential function to a vertex of the cube. We also show that, for every power of two $n \ge 2^{50}$, there exists a matrix $A \in \{-1,1\}^{n \times n}$ such that $\|Ax\|_\infty >1.0000002\sqrt{n}$ for every choice of signs $x \in \{-1,1\}^n$. The construction simply replaces a $2^{-22}$ fraction of the columns of a Hadamard matrix with independent random sign vectors. This is the first improvement over the $\sqrt{n}$ lower bound of Olson and Spencer (1978), which uses a Hadamard matrix.

The $\triangle$-convexity space on a finite, simple graph $G = (V,E)$ is the collection $\mathcal{C}$ of subsets $S \subseteq V(G)$ such that whenever $x \in V(G)$ forms a triangle with two vertices in $S$, we have $x \in S$. The members of $\mathcal{C}$ are called convex sets, and the convex hull of a set $S \subseteq V(G)$, denoted $\operatorname{Hull}(S)$, is the smallest member of $\mathcal{C}$ that contains $S$. The Carathéodory (resp., exchange) number, $c_{\triangle}(G)$ (resp., $e_{\triangle}(G)$), is the size of a largest Carathéodory (resp., exchange) independent subset of $V(G)$. It was shown by Anand et al. (JCMCC 126, 2025, 11–27) that $c_{\triangle}(G) \leq t(G)+1$ and $e_{\triangle}(G) \leq t(G) + 2$, where $t(G)$ is the number of triangles in $G$, and that these bounds are tight. They also computed $c_{\triangle}(G)$ and $e_{\triangle}(G)$ for a block graph $G$ in terms of the number and arrangement of non-$K_2$ blocks in $G$. In this paper, we point out a gap in the proof in Anand et al. of the first inequality, $c_{\triangle}(G) \leq t(G)+1$, which has consequences for the proof of the second inequality, $e_{\triangle}(G) \leq t(G) + 2$, as well. Moreover, the tightness results are inadvertently applied as characterizations of the extremal graphs, leading to incorrect computations of the Carathéodory and exchange numbers of block graphs in certain cases. We fix these gaps by giving a full proof of the first inequality via a different route from that in Anand et al. Together with the argument in Anand et al., this also completes the proof of the second inequality. Our proof also leads to a characterization of the extremal graphs for each bound, which we use to compute the Carathéodory and exchange numbers of block graphs and to identify the extremal block graphs. We also determine $e_{\triangle}(G)$ exactly for $k$-trees for every $k \geq 2$.

The principal matroid determinant $E_L$ of a linear space $L \subseteq \mathbb{P}^n$ has been introduced in recent work by Matsubara-Heo and Telen. In this paper, we give the complete factorization of this polynomial into its irreducible components, proving a conjecture of the aforementioned authors. Our methods rely on bounding local multiplicities and an étale-local description of the strata of reciprocal linear spaces developed by Elias, Proudfoot and Wakefield. We also discuss analogous questions for other coordinate-wise powers of linear spaces.

We study the temporal exploration problem on temporal digraphs. We prove that a lifetime of $O(n^2)$ suffices to guarantee the existence of a temporal exploration on always-unilateral temporal digraphs. We complement this with a $\Omega(n^2)$ lower bound, even in the case where each snapshot has maximum undirected degree 2; for always-strong temporal digraphs, the lower bound still holds even if the maximum undirected degree is 3. This stands in stark contrast with the undirected setting. For the large minimum degree setting, we show that a lifetime of $4n/3 - 1$ is sufficient and necessary for guaranteeing the existence of a temporal exploration on temporal digraphs where each snapshot is semicomplete. For always-strong temporal digraphs where each snapshot has minimum undirected degree at least $n - c - 1$, we prove that a lifetime of $O(cn)$ guarantees the existence of a temporal exploration, and we also prove that this is asymptotically tight. From a computational perspective, our results for temporal semicomplete digraphs also yield a polynomial-time, factor-$4/3$ algorithm for deciding if a temporal semicomplete digraph admits a temporal exploration within the first $\ell$ snapshots. We complement this showing that no polynomial-time, factor-$(4/3 - \epsilon)$ approximation algorithm exists, even if every snapshot is a tournament, unless P$=$NP.

For a finite simple graph $G = (V, E)$, we consider the quotient algebra $\mathcal M(G)$ of the polynomial ring in the variables $u_e$ (for $e \in E$) by the ideal generated by all products $u_e u_f$ for non-disjoint edges $e$ and $f$ (including all squares $u_e^2$). This quotient is called the matching algebra of $G$, since it has a basis indexed by the matchings of $G$. In this quotient, we define a subalgebra $\mathcal D(G)$ generated by the signed incidence sums $\theta_v=\sum_{e=(v,w)}u_e-\sum_{e=(w,v)}u_e$ for all $v \in V$ (where all edges of $G$ are oriented arbitrarily); we call this the Dunkl matching algebra. We show that, as a graded vector space, $\mathcal D(G)$ is dual to the span of all polynomials $p_M = \prod_{(i,j) \in M} (x_i - x_j)$, where $M$ ranges over all matchings of $G$. For the complete graph $K_n$, the latter span is a direct sum of two-row Specht modules (one in each degree); thus its Hilbert series is that of the Catalan triangle, and, in characteristic zero, the Dunkl matching algebra can be presented by linear and quadratic relations. For arbitrary graphs, we formulate the saturation problem of deciding when the selected matching Specht generators $p_M$ in a given degree $k$ span the full two-row Specht module $S^{(n-k,k)}$. We show that saturation in degree $k$ forces $k$-connectivity, that saturation in degree $2$ is equivalent to $2$-connectivity, and that $k$-linked graphs are saturated in degree $k$. We also prove saturation in every possible degree whenever the complement of $G$ is a matching. We show that the Dunkl matching algebra equals the full matching algebra exactly for forests, and give an explicit Hilbert series formula for unicyclic graphs.

In this paper, we investigate three fundamental convexity parameters of graphs under cycle convexity, namely the Helly number, Radon number, and rank. We first study the computational complexity of these parameters. For each of these parameters, we consider the associated threshold decision problem of determining whether the parameter of a given graph is at least a prescribed integer. We establish that all three problems are $\NP$-hard and $\W[1]$-hard when parameterized by the threshold. Moreover, we strengthen these results by showing that the $\NP$-hardness persists even when the input is restricted to planar graphs of maximum degree at most $6$. We also focus on the structural properties of connected graphs corresponding to extremal values of these parameters. In particular, we characterize the graph classes for which the three parameters attain the values $n-1$ and $n-2$, where $n$ is the order of $G$.

The limit set of Conway's Game of Life collects the configurations that can still appear at arbitrarily late times: those admitting predecessors of every finite depth. Answering a question of Salo and Törmä (ICALP 2022), we prove that two fixed finite patterns can each occur in the limit set, yet can never be found together in a single configuration of it, at any relative position; equivalently, the spatial translation action on the limit set is not topologically transitive. Our method is to make a persistent marker force a periodic lane to grow in sufficiently deep predecessors, until two perpendicular lanes are forced to intersect and prescribe incompatible values at a common cell. The argument uses two computer-verified local implications.