Papers

In 1992, Billera and Sturmfels introduced coherent monotone paths on polytopes as part of their description of the fiber polytope construction, and later in 1994 showed with Kapranov that these coherent monotone paths capture the topology of the space of all monotone paths, paths from a minimum to a maximum, in the directed graph of a polytope with orientation induced by a linear function. Those results motivate the following analog of the polynomial Hirsch conjecture: Does there always exist a coherent monotone path of polynomial length on a polytope for any choice of orientation induced by a linear function? We show this is not the case by exhibiting a family of polytopes and corresponding linear functions for which every coherent monotone path is exponentially long. As applications, we strengthen longstanding results pertaining to lower bounds for the shadow simplex method, geometric transversals in discrete geometry, and parametric linear optimization.

For normalized sequences $\left(g(n)\right)_{n\in\mathbb{N}}$ we consider recursively defined polynomials $P_n^g(x)$. In this paper we study their submultiplicative property, viewed as a Bessenrodt–Ono type inequality for the partition function, and provide an effective criterion for establishing it.

We provide a combinatorial characterization for pairs of order-theoretic trees with homeomorphic ray spaces, answering an open problem proposed by Kurkofka ad Pitz. This solution is inspired by the introduction of a transfinite topological game, which allows us to characterize not only ray spaces through the existence of winning strategies for one of the players, but also their homeomorphic classes. As applications of these results, we obtain a new topological characterization for graph-theoretic end spaces (thus obtaining yet another solution to a recently solved problem of Diestel), as well as for edge-end spaces and completely ultrametrizable spaces. We also introduce a generalization of the class of ray spaces (which is strict, as witnessed by the Sorgenfrey line). Furthermore, we establish that, for subspaces with cardinality less than continuum of end spaces, the scattered property is equivalent to the property of being, itself, an end space. At last, we determine that ray spaces in a couple of classes fail to have their product with any non-discrete space as a ray space.

Let $G$ be a two-orbit permutation group on $n > 2$ points. We show that $G$ contains either a derangement or an element of prime-power order with a unique fixed point. As a corollary, if the orbits of $G$ have length $n_1$ and $n_2$ and $\gcd(n_1, n_2-1) = \gcd(n_1-1, n_2) = 1$, then $G$ contains a derangement. The special case $n_1 = n_2$ was recently conjectured by Ellis and Harper and proved under various restrictive hypotheses. We prove our result by reducing to the case of simple groups and leveraging the classification of normal $2$-coverings of simple groups due to Bubboloni, Spiga, and Weigel.

We study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement $\max$, $+$, or multiplication with a positive constant. For such circuits, we prove exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. As a corollary, we obtain an exponential size separation between monotone and non-monotone maxout neural networks, which generalize the popularly used ReLU neural networks. One conclusion from this is that neural network models with enforced convexity constraints, such as input-convex neural networks (ICNNs), sometimes need to be exponentially larger than their unrestricted counterparts in order to express the same functions.

In this article, we discuss general generating functions for partitions of $n$, simultaneously $s$-regular and $t$-distinct in 3-colors. In addition, we obtain infinite families of congruences modulo powers of 3 for specific values of $(\ell,t)$. For instance, for positive integers $n$ and $k$, we have \begin{align*} \sum_{n=o}^{\infty}RD_3^{3,3}\left(3^kn+\frac{3^k+1}{2}\right)q^n\equiv0 \pmod{3^{k+1}}. \end{align*}

Aardal and Lenstra systematically studied hard knapsack problems of the form $a_1x_1+\cdots+a_nx_n=b$, where $a_i=p_iM+r_iN$, $(M,N)$ is a coprime pair of positive integers, and the integers $|p_i|, |r_i|$ are small relative to $M$ and $N$. We investigate the corresponding challenging denumerant problem (i.e., counting the number of nonnegative integer solutions) and present a polynomial-time algorithm. This eliminates the computational bottlenecks caused by large values of $M$, $N$ and $b$. The proposed algorithm achieves a time complexity of $O(n^4\Delta^2\log n\log\Delta)$, which depends solely on the parameters $n$ and $\Delta=\max_{i,j}|r_i p_j - r_j p_i|$. Moreover, we consider the problem of expressing a general vector $(a_1,\dots,a_n)$ in the above form using the LLL algorithm.

The toughness of a non-complete graph $G$, denoted $\tau(G)$, is defined as \[ \tau(G) = \min\left\{ \frac{|S|}{\omega(G-S)} : S \subseteq V(G),\ \omega(G-S) \geq 2 \right\}, \] where $\omega(G-S)$ is the number of components of $G - S$. For a complete graph $G$, we define $\tau(G) = \infty$. A graph $G$ is $t$-tough if $\tau(G) \geq t$. For a positive integer $k$, a graph $G$ is $(K_2 \cup kK_1)$-free if it contains no induced subgraph isomorphic to $K_2 \cup kK_1$. Recently, Liu \cite{liu} showed that every $2k$-connected $(K_2 \cup kK_1)$-free graph $G$ with $\tau(G) > 1$ is Hamilton-connected. In this paper, we strengthen this result by proving that every $(k+1)$-connected $(K_2 \cup kK_1)$-free graph $G$ with $\tau(G) > 1$ and minimum degree $\delta(G) \geq 2k$ is Hamilton-connected. Moreover, by imposing restrictions to the independence number $\alpha(G)$, we prove that every $k$-connected $(K_2 \cup kK_1)$-free graph $G$ of order $n$ with $2k+1 \leq \alpha(G) < \frac{n}{2}$ and $\delta(G) \geq 2k$ is Hamilton-connected, and that the bounds on $\alpha(G)$ are sharp.

Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on $\mathbb C^d\otimes\mathbb C^d$, obtained as the marginal over an environment $\mathbb C^s$ of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level $m\in\mathbb N$ in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension $s=\lambda_md^2$ at which the behavior of the associated random state switches from violating to satisfying it, with probability going to $1$ as $d$ grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension $d$ is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.

Pilatte recently proved that there exists an infinite Sidon set of positive integers which is an asymptotic basis of order $3$, answering a problem posed by Erdős, Sárkőzy and Sós in 1994. In this paper, we strengthen this result by proving that for any $0<\eta<0.0527$, there exists an infinite Sidon set $\mathcal{S}\subset \mathbb{N}$ which is an asymptotic basis of order $3-\eta$; that is, every sufficiently large integer $m$ can be represented as \[ m=s_1+s_2+s_3 \] for some $s_1,s_2,s_3\in \mathcal{S}$ satisfying \[ \min\{s_1,s_2,s_3\}\leq m^{1-\eta}. \] To prove this, we develop a truncated version of Pilatte's construction and use a deep result of Sawin on sums of Dirichlet convolutions of the von Mangoldt function over function fields.

Inspired by recent advances in notions of spectral approximation of digraphs [Ahm+20], we study spectral algorithms for finding periodic structures in digraphs via the spectrum of a class of rotated Laplacian matrices. This class of Laplacian matrices was previously studied by Lange, Liu, Peyerimhoff, and Post [Lan+15]. We consider a notion of periodicity ratio that generalizes the bipartiteness ratio of Trevisan [Tre09], and show that it is closely related to the spectrum of rotated Laplacian matrices. In particular, if the digraph is strongly connected and represents a Markov chain, this periodicity ratio for a given $p \in \mathbb{N}$ is a quantitative measure of how close this Markov chain is to having periodicity $p$. We propose and analyze a periodicity-ratio variant of the spectral algorithm by Louis, Raghavendra, Tetali and Vempala [Lou+12]. We show that the algorithm runs in randomized polynomial time and can find many nearly periodic components (i.e, components with small periodicity ratio). This also implies a new higher-order Cheeger-type inequality for periodicity in the spirit of that in [Lou+12; LOT14]. As part of our analysis, we prove a new theorem that upper bounds the probability that the largest magnitudes of two sequences of coordinate-wise correlated complex Gaussian random variables occur at different indices, which may be of independent interest. Previously, an analogous result was known only for real Gaussian random variables.

Phylogenetic networks are used to represent the evolutionary history of species. Due to biological interpretations and computational advantages, researchers have focused on restricted classes of phylogenetic networks, such as tree-child, orchard, and tree-based. These classes capture different notions of tree-likeness: tree-child networks require every internal vertex to have a taxon reachable by a tree path, orchard networks are trees with horizontal arcs (for modelling histories rife with horizontal gene transfers), and tree-based networks are trees with additional (not-necessarily horizontal) arcs. A natural question to ask is “how far is a given network from belonging to a particular class?” This motivates the study of proximity measures, which measure the minimum number of graph modifications required to transform a network into one belonging to a particular class. In this paper, we consider three proximity measures based on leaf addition, valid arc deletion, and arc deletion. We study pairwise comparability of the proximity measures, prove complexity results, and derive extremal bounds for the classes of tree, tree-child, orchard, and tree-based networks.

Wegner conjectured that every finite family $\mathcal R$ of axis-parallel rectangles satisfies $\tau(\mathcal R)\le 2\nu(\mathcal R)-1$, where $\nu$ is the packing number and $\tau$ is the piercing number. Ajwani, Gajjala, Raman, and Ray recently disproved this by constructing a triangle-free counterexample on $2196\cdot 8^9$ rectangles and, using a computer-assisted package-and-port recursion, obtained a standard LP gap of $17891/8064$ for Maximum Independent Set of Rectangles. We give a simpler and hand-checkable counterexample with $64$ rectangles. It is built from an eight-rectangle gadget whose independent sets inject into four ordered slots; we then use four horizontal and four vertical copies of this gadget to form a triangle-free family with $\nu=16$ and $\tau\ge 32$. We use the same horizontal-vertical step to define recursive families of rectangles $P_r$ with $\nu(P_r)=4^{2^r}$. For the standard clique, equivalently point, relaxation we obtain a finite gap $73/32$ at $P_3$, improving the previous benchmark of $17891/8064$. We then construct recursive fractional solutions and matching piercing sets showing $\lim_r \alpha^*(P_r)/\nu(P_r)=\lim_r \tau(P_r)/\nu(P_r)=5/2$. Finally, by disjoint union with isolated rectangles, we show that every rational $t\in[1,5/2)$ occurs as a standard LP gap and also as a packing-piercing ratio for suitable rectangle families.

In the present paper, we investigate two local zero-sum problems. Let $n,k\ge 2$. We denote by $\mathsf{D}^*(n,nk)$ (resp. $\eta^{*}(n,nk)$) the smallest positive integer $\ell$ (if exists) such that, from any given $\ell$ integers not divisible by $n$, one can select some (resp. at most $n$) of them whose sum is divisible by $n$ but not by $nk$. We prove that both $\mathsf{D}^*(n,nk)$ and $\eta^{*}(n,nk)$ are equal to $2n-1$ if $\mathrm{rad}(n) \mid \mathrm{rad}(k)$ and infinite otherwise. The corresponding inverse problem is also determined. We denote by $\mathsf{D}_n^{\times}$ (resp. $\eta_n^{\times}$) the smallest positive integer $\ell$ such that, from any given $\ell$ integers coprime to $n$, one can select some (resp. at most $n$) of them whose sum $\sigma$ satisfies $\gcd(\sigma, n^2)=n$. We prove that $\mathsf{D}_n^{\times}=\eta_n^{\times}=2n-1$ if $n$ is a prime power, and determine its inverse problem.

We nearly settle a natural extremal question about set systems over $[n]$: the tradeoff between the {size} (number of sets) and the number of {full chains}. This question was initially raised by Johnson, Leader, and Russell [Combin.~Probab.~Comp., 2015] as a counterpart to Sperner-type results in combinatorics. Recently, a framework introduced by Ameli, Nederlof, and Wang, and independently by Dallant and Kozma [FOCS 2026] linked this question to the space- and time-complexity of Bellman-Held-Karp-style dynamic programming algorithms for permutation problems such as the traveling salesman (TSP). Precisely, they showed that a space-time product $\gamma^{n+o(n)}$ is feasible for the TSP, whenever a set system of (normalized) size $S$ and chain density $D$ exists, with $ \gamma = S^2/D$. In this paper we show an essentially {optimal} bound of $\gamma \approx 3.1819$ for this quantity, closing the gap between the previous best lower and upper bounds of $\gamma \geq 3.015$ and $ \gamma \leq 3.572$ respectively. This implies a TSP algorithm with space-time product $O(3.1819^n)$ for input size $n$, as well as a limit to further improvements in this broad framework. More generally, we can obtain close to optimal values $D$ for any feasible value $S$, effectively settling the question of the number of full chains at every size. The crucial step towards our results is casting the extremal combinatorics question as an {information~vs.~entropy} tradeoff involving two random variables. This reformulation {exactly} captures the optimal tradeoff for the combinatorial problem, leading to a framework in which primal-dual certificates can be derived, proving rigorous upper and lower bounds on $\gamma$. We also give a further application of our techniques, improving a bound of Duffus, Sands, and Winkler on the minimum size of fibres in the Boolean lattice.

We consider the following additive Latin transversal problem. Given a multiset $A=(a_1,\dots,a_k)$ of elements of $\mathbb Z_m$ and a set $B\subseteq\mathbb Z_m$ of cardinality $k$, the task is to order $B$ as $b_1,\dots,b_k$ so that the sums $a_i+b_i$ are pairwise distinct. When $k=m$, Hall proved that a solution exists if and only if $\sum_{i=1}^m a_i\equiv 0 \pmod m$; moreover, his theorem yields a polynomial-time construction. Alon proved that a solution always exists when $m$ is prime and $k<m$, but no polynomial-time construction is known in general. Our main algorithmic contribution is a direct randomized algorithm for Color-Counted Matching: given an edge-colored graph and prescribed target counts for the colors, find a matching using exactly the prescribed number of edges of each color. If $q$ is the sum of the target counts and $h$ is the number of colors, our base-$(q+1)$ reduction to Exact Red Matching, combined with the algorithm of Mulmuley-Vazirani-Vazirani, gives a randomized algorithm with running time $\left(|V|^2+|E|(q+1)^{h-1}\right)^{O(1)} $ for an input graph $(V,E)$. Thus the dependence on the target matching size is $q^{O(h)}$, up to polynomial factors in the graph size. In contrast, applying the general matching-ILP theorem of Lassota and Ligthart as a black box yields a $q^{O(h^2)}$ dependence for the corresponding fixed-size color-counted instances. Applying this primitive to additive Latin transversals with $s=|\operatorname{supp}(A)|$, we obtain an algorithm in randomized time $(k+\log m)^{O(s)}$. In particular, additive Latin transversals are randomized polynomial-time constructible for every fixed support size.

Given a fence poset P , we define a new statistic on permutations, denoted by blP, that provides a combinatorial interpretation of the coefficients of the order polynomial of P , answering a question of Ferroni, Morales, and Panova (2025). Using the fact that the base polytope of a lattice path matroid can be decomposed into order polytopes of fence posets, we also obtain a combinatorial interpretation of the coefficients of the Ehrhart polynomial of the base polytope of Schubert matroids, answering a question of Stanley (1999). As an application of this statistic, we establish the first nontrivial lower bound for the linear coefficient of the Ehrhart polynomial of an order polytope. Finally, we conjecture generalizations of this statistic to skew-shape posets and circular fence posets.

In this paper we introduce the group action-stabilizer graph $GAS(G)$ of a group $G$ on a set $X$ with vertex set as the collection of all the group actions of $G$ on $X$, and any two vertices $\phi$ and $\psi$ are adjacent if and only if the non-trivial subgroups $\cap G_x^\phi$ and $\cap G_x^\psi$ of $G$ intersect non-trivially, where $G_x^\phi$ and $G_x^\psi$ are two stabilizers of $x$ with respect to the actions $\phi$ and $\psi$, respectively. We characterize a special subgraph $gas(G)$ of $GAS(G)$ in which the vertex set contains the actions $\phi$ of $G$ on $X$ such that $\cap G_x^\phi$'s are distinct. We determine the number of group actions within some specific groups and find certain conditions under which $GAS(G)$ is equal to $gas(G)$. We also examine the conditions under which $gas(G)$ and its complement are derived graph for a finite nilpotent group $G$.

We study the PBW filtration on covariant representations for the Lie superalgebra $\mathfrak{gl}(m|n)$. We prove for all covariant weights of the form $(\lambda|\mu,0^{n-1})$, that there exists a lattice polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence, we obtain degenerations of partial flag supervarieties for the supergroup $GL(m|n)$ into toric supervarieties.

Let $\mathscr{G}_{n,k}$ denote the family of all connected graphs of order $n$ with matching number $k$. Liu, Lou, and Trevisan~(Linear Algebra Appl., 2026) posed the following problem: Determine the spectrally minimal graphs in $\mathscr{G}_{n,k}$. In this paper we prove that for every graph $G \in \mathscr{G}_{n,k}$, $ \rho(G) \ge \sqrt{\frac{n + 2k - 3}{k}}, $ and we completely characterize the extremal graphs when $k \mid (n-3)$. As applications, we establish $\rho(G) + k \ge 3\sqrt[3]{n/4}$ for $k \ge 2$, settling the asymptotic order of $\rho + k$ as $\Theta(n^{1/3})$ – strictly smaller than the $\Theta(\sqrt{n})$ order suggested by the disproved Aouchiche–Hansen conjecture.

We prove an $\mathrm{MSO}_2$ zero-one law for a very sparse Erdős-Rényi graph after pruning by component order. Let $p_n=c_n/n$, where $c_n\to0$, and delete every component of order less than $f(n)$, where $f(n)\to\infty$. If \[ f(n)\bigl(\log f(n)+\log(1/c_n)\bigr)=o(\log n), \] then the resulting graph satisfies a zero-one law for $\mathrm{MSO}_2$, with quantification over sets of vertices and sets of edges. The proof combines uniform component counts, an MSO Feferman-Vaught decomposition for disjoint unions, and semilinearity of the order spectra of MSO-definable classes of finite trees. We also show that the term $f(n)\log f(n)$ cannot simply be omitted: star components can occur at first-order-visible Poisson thresholds. We further establish first-order limit laws for bond percolation on the discrete torus $T_L^d$. In the two-sided subpolynomial regime, pruning below a sufficiently slow threshold yields a zero-one law. For the unpruned model in either one-sided polynomial regime, the reciprocal exponents $\alpha=1/k$ are precisely the critical scales. At such a scale, an extended limit of $N p_N^k$ or $N q_N^k$ equal to $0$ or $\infty$ gives a zero-one law; a positive finite limit gives a convergence law but not a zero-one law; and the absence of an extended limit gives failure of convergence. Finally, $\mathrm{MSO}_1$ already detects the parity of the torus side length through bipartiteness, producing a natural obstruction to monadic convergence in a near-deterministic regime.

We give a tight single-change covering design with $v=26$ and $k=6$. This answers Problem 1 of the Nineteenth British Combinatorial Conference, which asked whether such a design exists with block size greater than $5$. We also describe the satisfiability search that found the design, including negative search results at the smallest admissible order $v=21$.

Using elementary $q$-series manipulations, we establish a positivity property for the tails of the Jacobi theta series. Specifically, for integers $k\ge 1$ and $n\ge 0$, define \[ \sum_{n\ge0}\sum_{m\in\mathbb{Z}}J_{k,n}(m)z^m q^{n} = \frac{(-1)^k q^{-\binom{k+1}{2}}}{(z)_{\infty}(q/z)_\infty} \sum_{j\ge k}(-1)^jq^{\binom{j+1}{2}}z^{-j}(1-z^{2j+1}), \] where $(a)_\infty:=\prod_{n\ge0}(1-aq^n)$ denotes the $q$-shifted factorial. We prove that for all integers $k\ge 1$ and $n\ge 0$, the coefficients $J_{k,n}(m)$ are positive for all integers $-(k+n)\le m\le k+n$.

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), the $\ell$-degree Turán density $\gamma^{(k)}_\ell(\mathcal{F})$ of $\mathcal{F}$ is defined as the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In this paper, we prove that for all $k>\ell>k/2$, every rational number $\alpha\in[0,1)$ can be realized as the $\ell$-degree Turán density $\gamma^{(k)}_\ell(\mathcal{F})$ for some finite family $\mathcal{F}$ of $k$-graphs. Furthermore, for any $k>\ell\ge 1$, we construct an explicit infinite sequence of values realized by single forbidden $k$-graphs: for each integer $q\ge2$, there exists a $k$-graph $F$ such that $\gamma^{(k)}_\ell(F)=1-q^{\ell-k}$. We also establish a strengthened non-principality property for $\ell$-degree Turán densities: for all $k>\ell>1$, there exist two $k$-graphs $F_1$ and $F_2$ satisfying $0< \gamma^{(k)}_{\ell}(\{F_1,F_2\})<\min\{ \gamma^{(k)}_\ell(F_1),\gamma^{(k)}_\ell(F_2)\}$.

In arXiv:2511.13287, we derived exact formulas for generating functions counting the number of $n$-ary words avoiding strictly increasing subwords of length $k$, and provided applications in probability theory as well as the continuous limit as $n\to\infty$. We also conjectured several corresponding formulas for the case where the “strictly” requirement is dropped. In this paper, we prove those formulas.

Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index $\chi'(G)$. A graph $G$ is called edge-chromatic $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)< \chi'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\Delta(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the influential Overfull Conjecture: If $G$ is a simple graph with $\Delta(G)>\frac{|V(G)|}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$. Motivated by the structural analysis for $4$-critical graphs (SIAM J. Discrete Math. 2019), we show more properties in this paper, especially four new forbidden configurations in any $4$-critical graph, and provide a new structural proof of Overfull Conjecture for graphs with maximum degree $4$.

Monadic dependence is a proposed structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A graph class is monadically dependent if the class of all graphs cannot be interpreted in its vertex-colored members using a fixed first-order formula. We prove two structural consequences of monadic dependence. First, every monadically dependent class has almost linear neighborhood complexity: for every graph $G$ in the class and every set $A\subseteq V(G)$, the family $\{N_G(v)\cap A : v\in V(G)\}$ has size $|A|^{1+o(1)}$. Second, every $n$-vertex graph in a monadically dependent class has radius-1 merge-width $n^{o(1)}$. Here, merge-width is the decomposition parameter of Dreier and Toruńczyk based on construction sequences; its radius-$r$ version measures local reachability among parts through already resolved pairs. This settles the radius-1 case of the conjectured connection between monadic dependence and almost bounded merge-width and provides the first decomposition-based structural description of monadically dependent graph classes. Our proof is algorithmic: we give an $\mathcal{O}(n^5)$-time algorithm that, given an $n$-vertex graph $G$ such that $|\{N_G(v)\cap A : v\in V(G)\}|\le O(|A|^d)$ for every $A\subseteq V(G)$, computes a construction sequence witnessing radius-1 merge-width $\mathcal{O}(n^{1-1/d}\log n)$.

A graph class is $k$-WQO if its $k$-labeled graphs are well-quasi-ordered under label-preserving induced subgraph embeddings. We show that every hereditary graph class that is $2$-WQO has bounded clique-width. Combined with the recent result of Dumas and Lopez, this confirms a long-standing conjecture of Pouzet: A hereditary graph class is $2$-WQO if and only if it is $k$-WQO for all $k\geq 2$, if and only if it is $\forall$-WQO, that is, its labeled graphs are well-quasi-ordered for every possible well-quasi-ordered label set. Our proof builds on a recent structure/non-structure dichotomy for the model theoretic notion of monadic dependence by Dreier, Mählmann, and Toruńczyk. Through the non-structure characterization by forbidden induced subgraphs, we show that every hereditary $2$-WQO graph class is monadically dependent. Leveraging the Ramsey-theoretic structural properties provided by monadic dependence, we then establish bounded clique-width by ruling out the existence of large well-linked sets, which are the canonical obstructions for clique-width.

We investigate the combinatorial structure of the tropical characteristic polynomial of symmetric matrices using the tropical permanents of their principal submatrices. We establish new inequalities for the leading coefficients of the tropical characteristic polynomial, revealing concavity properties of the coefficient sequence and yielding necessary conditions for a sequence to arise as the coefficient sequence of the tropical characteristic polynomial of a symmetric matrix. These results provide a deeper understanding of the structure of tropical characteristic polynomials associated with symmetric matrices.

We examine combinatorial structures that generalize $(k,n)$-nets, orthogonal arrays, and mutually orthogonal Latin squares. By a reticulation we mean a point set and two collections (types) of families of lines such that two lines of different types meet in exactly one point and each family of lines partitions the point set. The number of points incident with any line depends only upon the type of the line, and every point is incident with the same number of lines of a given type. Each choice of a single line family of each type leads to an arrangement of the points into a rectangular grid. Recording the line containing a given point in the corresponding position of an array gives a generalization of sets of mutually orthogonal Latin squares, dubbed a cooperative system. A cooperative system consists of a collection of column-Latin matrices and a collection of row-Latin matrices such that each column-Latin matrix is orthogonal to each row-Latin matrix. Recording lines which contain a given point as a tuple gives a generalization of certain orthogonal arrays, dubbed semi-orthogonal arrays. Notions of parastrophy and isotopy for cooperative systems, corresponding to permuting families of lines and lines within each family, are introduced. Constructions of small reticulations and three recursive constructions of larger reticulations are given.