Papers

This study investigates the diametral metric dimension of generalized corona graphs, where the main graph is connected graph and the branch graphs form a sequence of connected graphs. The concept of diametral metric dimension is an extension of the metric dimension concept, requiring the resolving set to contain all diametral vertices. To determine the diametral metric dimension of a generalized corona graph, one must first identify the distance of each vertex in the graph and the graph's diametral set. Subsequently, a resolving set containing the diametral set is determined. The results show that the diametral metric dimension of the generalized corona graph depends on the diameter of the main graph and the metric dimensions of the graphs in the sequence. These concepts provide theoretical insights into resolving structures in the planning infrastructure and motivate further studies on other graph families and graph operations.

The weighted $k$-matroid intersection problem asks for a maximum-weight set that is independent in each of $k$ matroids on a common ground set. The natural LP relaxation optimizes over the intersection of the $k$ matroid independent set polytopes. It is conjectured that this LP has integrality gap at most $k-1$. The conjecture is known for $k\le3$, but for $k\ge4$ the best general upper bound was $k$. We improve this bound to $k-1+1/k$. More generally, we prove that the natural LP of a $p$-matchoid has integrality gap at most $p-1+1/p$, with a deterministic LP-relative algorithm attaining the same factor. The matchoid extension resolves the $p$-matchoid part of a conjecture of Lee, Sviridenko, and Vondrák; projective planes give explicit tight instances whenever one of order $p-1$ exists.

We give a construction of admissible partitions that yields the Bernoulli decomposition theorem

Generalized staircase partitions are obtained by replacing each box of an ordinary staircase with a fixed rectangle. We determine the unique shortest horizontal-strip sequence between consecutive generalized staircases and its conjugate vertical-strip sequence. For monic Macdonald polynomials, we derive explicit finite principal-specialization ratios along both sequences. For arbitrary nested partitions, coefficientwise nonnegativity after removal of the monomial factor forces independence of $q$. Along the horizontal sequence, this nonnegativity is characterized by rectangular complementation, apart from the one-column case. Along the vertical sequence, it occurs at the smallest admissible number of variables, apart from the initial column case. Endpoint ratios yield a triangular product formula and a centered product with exchange, reciprocity, and inversion identities. Hall-Littlewood, Jack, and Schur specializations give Gaussian-polynomial, finite-product, and tableau formulas, respectively.

We construct an explicit involution on Dyck paths satisfying $\defc\le\min(\area,\dinv)$ that interchanges area and dinv, where $\defc=\binom n2-\area-\dinv$ and $n$ is the semilength. In addition, we give an involution on Dyck paths with $\defc \le 2n-8$ that interchanges area and dinv. Finally, we give an explicit partition formula for the portion of the $q,t$-Catalan polynomial of total degree at least $\binom n2-2n+8$, thereby proving a conjecture of Lee and Li~\cite[Conjecture~4]{LeeLi11}. Altogether our results give a combinatorial explanation of $q,t$-Catalan symmetry in the region $\defc\le\max(2n-8,\min(\area,\dinv))$. Our constructions rely on a number of new combinatorial objects developed here and three main external tools: the dual Dyck insertion of \cite{Hawkes26}, the Garsia–Milne involution principle~\cite{GarsiaMilne81},~\cite{Doyle19}, and the partition bijection of ~\cite{LoehrWarrington09}.

A graph $G$ is deletion-induced-saturated for $H$ if $G$ has an edge, contains no induced copy of $H$, and deleting any edge of $G$ creates an induced copy of $H$. We prove, with finite certificate verification, that a finite graph $H$ admits such a finite graph $G$ if and only if $H$ is not complete. This resolves the deletion conjecture of Fan, Hajebi, Hajebi and Spirkl. The main step transfers suitable free amalgamations to finite extensions using a local lifting theorem of Auinger, Bitterlich and Otto. A second criterion treats edge addition by protecting specified nonedges and then taking a maximal induced-$H$-free completion. Structural results of Bonamy, Groenland, Johnston, Morrison and Scott reduce the remaining targets to dense templates and a finite hereditary class. Two uniform constructions in halved cubes handle the dense templates. The finite part is supported by exhaustive coverage certificates, structural certificates and explicit hosts, including a circulant graph on $30$ vertices.

Mitra, Offen, and Sayag introduced distinguished multisegments and multisegments of Speh type, and proposed that every distinguished multisegment is of Speh type, together with a related duality hypothesis (see A. Mitra, O. Offen, and E. Sayag, Klyachko Models for Ladder Representations, Documenta Math. 22 (2017), 611-657). They proved these statements for sets of segments and when at most two segments share an endpoint. We develop a combinatorial theory of relevant decompositions and prove that relevance is preserved under the involution $\Delta\mapsto\Delta^\vee$ together with reversal of standard order. Hence, $\mathfrak m$ is distinguished if and only if $\mathfrak m^\vee$ is distinguished, so the two hypotheses are equivalent. We define $S_{\mathfrak m}(\Delta)=\sum_{t\ge0}(-1)^t\mathfrak m(\nu^t\Delta)$ and show that $\mathfrak m$ is of Speh type if and only if $S_{\mathfrak m}(\Delta)\ge0$ for every segment $\Delta$. Using rigidity properties, we obtain a peeling theorem and a numerical tameness condition, and prove both hypotheses for every tame multisegment, extending the previously known classes. Finally, we construct a five-segment multisegment that is not distinguished, although every standard order with non-increasing endpoints admits a non-trivial relevant decomposition. Thus, endpoint-ordered witnesses alone cannot prove the hypothesis in general. The remaining case reduces to multisegments for which both $\mathfrak m$ and $\mathfrak m^\vee$ are non-tame.

An induced matching is a matching that forms an induced subgraph. A graph is $t$-color-critical if removing some induced matching of size $t$ lowers its chromatic number, but removing any $t-1$ vertices does not. Let $F$ be a $t$-color-critical graph with $\chi(F)=r+1$. For sufficiently large $n$, Simonovits determined the unique edge-extremal $F$-free graph on $n$ vertices. Recently, Zheng, Li and Li [Linear Algebra Appl.\ 730 (2026) 546–565] conjectured that, for $t\ge 2$ and $r\ge 3$, the join $K_{t-1}\vee T_{n-t+1,r}$ uniquely maximizes the signless Laplacian spectral radius among all $n$-vertex $F$-free graphs when $n$ is sufficiently large. In this paper, we prove this conjecture. In contrast to the usual spectral arguments, our proof of this conjecture relies on two techniques of a rather different flavour. Our first technique is an analogue of Zykov symmetrization for the signless Laplacian matrix. Our second technique is an induction on $n$, from which we obtain the lower bound on the smallest entry of the Perron vector of a signless Laplacian spectral extremal graph rather than a structural statement.

For all positive integers $\ell$ and $r$, we prove that a finite simple rank-$r$ gammoid with no $U_{2,\ell+2}$-minor has at most $\ell(r-1)+1$ elements. This gives an affirmative answer to Problem~7.2 of Boretsky and Walsh [European J. Combin.\ 137 (2026), 104419], for both gammoids and transversal matroids. The bound is sharp for every $\ell$ and $r$. The key estimate is that, for every finite simple gammoid $M$, the sum of $|L|-2$ over its long lines is at most $|E(M)|-r(M)$.

Let $G$ be a Borel digraph with maximum out-degree $d \in \mathbb{N}$. We show that $G$ admits a random Borel $(2d+1)$-coloring for which every edge is almost surely not monochromatic. This gives a simpler proof of a recent result of Pelayo-Gómez: such a graph $G$ admits a measurable proper $(2d+1)$-coloring with respect to any Borel probability measure on $V(G)$. Our proof is an adaptation of Pelayo-Gómez's proof to the randomized Borel setting.

If $U$ is a minimal, diskbusting, finite list of words in a free group $F_n$ of rank $n$ such that the sum of the lengths of words in $U$ is at most $2n+4$, we prove that the natural presentation complex of the Baumslag double of $F_n$ along $U$ virtually contains a $\pi_1$-injective embedded closed hyperbolic surface. This verifies the Tiling Conjecture of Kim and Wilton for this type of lists of words, and in particular, implies that the corresponding Baumslag double contains a hyperbolic surface subgroup.

We present a simplified proof of the Erdős-Sós Conjecture on trees in graphs, which is based on its recent resolution by GPT-6 Astra. At the same time, we also give another proof of the corresponding conjecture for antidirected trees in digraphs, posed by Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé. Our proof uses the language of random cyclic orderings in order to take advantage of symmetries that were obscured in the original argument, and we hope that the reader will find it intuitive.

For a $k$-uniform hypergraph $\mathcal H$, the shadow of $\mathcal H$ is the graph whose edges are the pairs covered by a hyperedge. In this paper, for all sufficiently large $n$, we determine the $n$-vertex $k$-uniform hypergraphs of maximum adjacency-tensor spectral radius whose shadow has no $K_t$ minor, for every $t\ge k+1$, and those whose shadow has no $K_{s,t}$ minor, for every $2\le s\le t$ with $s+t\ge k+1$ and every residue of $n-s+1$ modulo $t$; outside these ranges the problems are trivial. This extends to uniform hypergraphs the theorem of Tait on graphs with no $K_r$ or $K_{s,t}$ minor, whose remaining residues were settled by Zhai and Lin. In each case the extremal hypergraph is unique, and it is the $k$-clique hypergraph of the join of a clique with a graph that we call the light part. For $K_{s,t}$ the answer depends on $j=k-s+1$. When $j\le1$, the maximum has order $n^{(k-1)/k}$, and the light part is the one found by Zhai and Lin for the adjacency matrix, including its exceptional components. When $j\ge2$, a regime that does not occur for graphs, the maximum has order $n^{(s-1)/k}$ and $t$ enters its leading constant. The light part then consists of copies of $K_t$ and one smaller clique, with a single exception: for $(k,s,t)=(9,8,8)$ and $n-s+1\equiv2\pmod 8$, the complement of the Petersen graph appears. When the smaller clique has between $1$ and $j-1$ vertices, the extremal graph is not unique. In particular, for $t=8$, $4\le s\le7$, $k=s+1$ and $n-s+1\equiv2\pmod 8$, the clique hypergraph of the extremal graph of Zhai and Lin is not extremal. For $j\ge2$ the light part is determined by a weighted clique inequality, which for $j\ge3$ follows from a weighted form of the closed-neighborhood counting of Chao and Dong.

We address a problem on large signed sums of unit vectors that arose in work of Brugger, Fiedler, González Merino and Kirschbaum and was later formulated in its present form by Ambrus and Nietert. Given $d+1$ unit vectors $u_1,\ldots,u_{d+1}$ in $\mathbb R^d$, with $d\ge2$, the problem asks for the smallest possible value of \[ \max_{\varepsilon_i=\pm1} \left\|\sum_{i=1}^{d+1}\varepsilon_i u_i\right\|. \] We prove that this value is $\sqrt{d+2}$. We also determine all equality cases: up to independent sign changes and orthogonal transformations, they consist of the vertices of a centered regular simplex of positive even dimension together with an orthonormal basis of its orthogonal complement.

The permanent and mixed discriminant of positive matrices are classic problems for which we do not expect an efficient algorithm for exact computation. Thus much work has been done to understand how well we can bound and approximately compute these quantities. One line of research in this area begins with the results of the first author, where van der Waerden lower bounds of $\frac{n!}{n^n}$ are proven for doubly stochastic inputs for both problems, using a simple proof via stable polynomials. Along with the bound itself, the same techniques are used to show that the permanent and mixed discriminant are uniquely minimized at a certain natural symmetric input. In this paper, we generalize those results in two ways. First, we extend the unique minimization results beyond doubly stochastic inputs to other marginals which are near doubly stochastic. This yields the first such unique minimization results for the mixed discriminant beyond the doubly stochastic case. We also discuss why one cannot hope similar results to hold in general for all marginals. Second, we extend the unique minimization result for real stable polynomials to strongly log-concave (aka Lorentzian) polynomials in the doubly stochastic case. This captures an analogous previous result on unique minimization for the mixed volume. Finally, we discuss various open problems related to these results.

We prove that every finite Helly complex is isomorphic, at the level of its $1$-skeleton, to the Hellyfication (equivalently, the discrete injective hull) of its combinatorial boundary equipped with the metric induced from the complex. In particular, the boundary-rigidity phenomenon for Helly complexes proved by Blufstein-Chalopin-Chepoi admits a canonical injective-hull interpretation, independent of reconstruction procedures based on dismantling.

Every finite planar set of at least $880$ points contains four collinear points or six pairwise visible points.

We give an elementary proof of the Komlós conjecture by simplifying the recent proof of Guo, Fang, and Lu. We show that any vectors $v_1,\ldots,v_n\in\mathbb{R}^d$ with $\|v_i\|_2\le1$ admit signs $\varepsilon_i\in\{-1,1\}$ such that $\|\sum_{i=1}^n\varepsilon_i v_i\|_\infty\le36$. The proof uses only elementary combinatorial and probabilistic arguments and basic calculus.

We study frieze patterns over the integers that are allowed to have wild entries. We introduce the quiddity number as a new invariant. The quiddity number is then used to classify strongly connected components of the directed graph $\Gamma_{2,n}(\mathbb{Z})$. Furthermore, we show that every finite simple directed graph arises as an induced subgraph of a directed graph $\Gamma_{2,n}(\mathbb{Z})$ for $n$ sufficiently large.

We prove that the independence sequence of every sufficiently large forest is unimodal. The result follows from establishing log-concavity on a central interval of the sequence, along with monotonicity of the initial and final segments. The main analytic step is a central limit theorem for the size of a random independent set sampled with the hard-core model, uniform over all forests and over an interval of positive fugacities.

Condensed configurations are compact incidence data obtained by grouping the cyclic flats of a matroid. We show that their inverse incidence matrices give explicit Schubert expansions and hence determine every valuative or covaluative matroid invariant. For the extended binary Golay matroid, this unexpectedly produces non-real-rooted Kazhdan-Lusztig and Z-polynomials. To our knowledge, the latter is the first counterexample in the literature.

A polyhedron is a planar, $3$-connected graph. We iteratively construct all polyhedra (save for pyramids) from a unique starting graph, namely the square pyramid, via two graph transformations. This builds upon a previous construction, that starts from the full class of pyramids, and applies the same transformations. In a related result, we iteratively construct all quadrangulations of the sphere where all $4$-cycles are facial, i.e., the class of radial graphs of the polyhedra (save for antibipyramids), from a unique starting graph, namely the square antibipyramid, via a unique graph transformation. This builds upon a previous construction, that starts from the full class of antibipyramids, and applies the same transformation.

We prove new small-ball inequalities for boolean matrix-series. The leading example is $\mathbb E_s[{\text{det}(I-S^2)^\beta\,\mathbf 1_{\{\|S\|<1\}}}]\ge e^{-O(\beta\tau)}$, which holds for boolean matrix-series $S=\sum_i s_iA_i$ formed using symmetric matrices $A_1,\dots,A_n$ and uniformly random signs $s\in\{\pm1\}^n$. Specifically, this inequality holds for all $\beta\ge1$ with $\tau=\sum_i\text{Tr} A_i^2$, as soon as the maximum of $(\text{Tr} A_i^2)_{i=1}^n$ and a certain variance term are bounded above by universal constants. The proof combines the Gaussian reciprocal estimate of (Akbas and Sra 2026), the directional-variation signing theorem of (Guo, Fang, and Lu 2026), and a replica argument that turns existence into a Gibbs law on good signings. Most notably, boolean small-ball delivers a new, interlacing-free proof of Kadison-Singer (most general case); it also recovers Matrix Spencer and Komlós as quick corollaries, while yielding more than six almost immediate proofs of an assortment of discrepancy theoretic problems.

Gowers, Green, Manners, and Tao (Annals '25) recently resolved Marton's polynomial Freiman-Ruzsa conjecture. We give an algorithmic counterpart to their result: given uniform sampling and membership-oracle access to a set $A \subseteq \mathbb{F}_2^n$ with doubling constant at most $K$, our algorithm outputs a subspace of size at most $|A|$ whose $K^{O(1)}$ translates cover $A$. The algorithm runs in $\textsf{poly}(n,K)$ time. As applications, we obtain polynomial-time algorithms for a variety of learning problems, including quadratic Goldreich-Levin, improper agnostic tomography of stabilizer states, and tomography of quantum states with bounded stabilizer extent.

We introduce a new method that brings the combinatorics of hyperplane arrangements into the study of representation zeta functions of compact Lie groups. For the Witten zeta function $\zeta_\Phi(s)$ associated with a root system $\Phi$, our method yields elegant formulas for $\zeta_\Phi(0)$ and $\zeta_\Phi'(0)$ in terms of the exponents of various parabolic subsystems of $\Phi$. Such formulas do not appear to be readily accessible through the conventional analytic techniques in the literature. More generally, the method applies to a broad family of conical zeta functions, expressing these two special values through the Möbius function of the intersection poset of the associated hyperplane arrangement.

We prove that for every acyclic matrix pattern $P$ it holds that $\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$.

A longstanding conjecture attributed to Smith (1984) asserts that for every $k\ge2$, any two longest cycles in a $k$-connected graph share at least $k$ vertices. In this paper, we prove the first linear lower bound, showing that any two longest cycles in a $k$-connected graph share at least $k/600$ vertices. Departing from previous Turán-type extremal arguments, we develop a novel structural approach that also yields applications to related problems on longest cycles and paths.

We give an FPRAS for the permanent of an $n\times n$ $0/1$ matrix with running time $\widetilde{O}(n^{3.5}\varepsilon^{-2})$. Our algorithm extends to a strongly polynomial FPRAS for arbitrary nonnegative matrices, as in previous works. Jerrum, Sinclair, and Vigoda (2004) gave the first FPRAS for the permanent of a nonnegative matrix. The running time was subsequently improved to $\widetilde{O}(n^7)$ by Bezáková, Štefankovič, Vazirani, and Vigoda (2008), and recently to $\widetilde{O}(n^6)$ by Chen, Vigoda, and Yang (2026). We introduce a multicommodity-flow bound inspired by electrical flows, replacing the usual path-length factor by routing energy. For a boosted version of the classical JSV chain, we prove a relaxation-time bound of $O(n^3\log n)$ and show that stationary trajectories of this length estimate all stationary hole-pattern probabilities, yielding an $\widetilde O(n^5)$-time FPRAS algorithm. Our new hole-weighted slide (HWS) chain improves both bounds to $O(n^2\log n)$, yielding an $\widetilde O(n^4)$-time algorithm. Finally, we obtain the claimed $\widetilde O(n^{3.5})$ running time by using a subset of $\widetilde{O}(\sqrt{n})$ checkpoint temperatures in an iterated sequence of warm-starts to obtain initializations at every temperature.

Can one construct a graph $G$ on the set of integer points ${\mathbb Z}^2$ in the plane such that the length of the shortest path between any two vertices of $G$ differs from their Euclidean distance by at most an absolute constant? This question of Benjamini, Erd\H os, Kleiner, Kozma, Schramm, and the first-named author has been open for a long time. We give an affirmative answer to a weaker form of this question, based on the following geometric statement, which is of independent interest. There exists a constant $c>0$ such that for every $i=1,2,\ldots,$ every $\rho$-fat plane convex set $S$ can be cut into $2^i$ convex pieces of equal area, each of which is at least $c\rho$-fat. (A convex set is $\rho$-fat if the ratio of its inradius to its circumradius is at least $\rho$.) We prove that there exists an (unweighted) spanning subgraph $G$ of an enlarged copy of ${\mathbb Z}^2$ such that, for every pair of vertices at Euclidean distance $d$, their shortest-path distance in $G$ lies between $d-O(1)$ and $d+o(d^{5/6})$. The same bound can be achieved by a planar graph with vertex set ${\mathbb Z}^2$, in which every edge joins two vertices at Euclidean distance at most 2.

Godsil, Roberson, Šámal and Severini conjectured that the quantum chromatic number of the categorical product of two graphs equals the minimum of the quantum chromatic numbers of the factors. We disprove this conjecture: we construct explicit finite graphs $G,H$ with \[ \chi(G\times H) \leq 1538 < 1539 = \min(\chi_q(G),\chi_q(H)).\]The graphs are obtained from Zhu's counterexample to Hedetniemi's conjecture by using a base graph for which the Lovász theta number of the complement, and not only the fractional chromatic number, is large. The lower bound for the first factor is the theta bound. For the second factor we adapt Zhu's argument to projections that do not commute: the step that fixes the colors of a clique is replaced by identities between operators. Both lower bounds hold for colorings by projections in an arbitrary nonzero unital $C^*$-algebra. Hence the conjecture also fails for the spatial, approximate, commuting-operator and $C^*$-algebraic variants of the quantum chromatic number. We also give smaller counterexamples certified by exact integer data. The graph constructions, the certificates and the counterexample statements in the projective formulation are formalized in Lean~4.