The \emph{minimum positive codegree} $δ^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The \emph{positive codegree Turán density} of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $δ^+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0<γ^+(F_1, F_2) < \min\{γ^+(F_1), γ^+(F_2)\}. $$

We prove that every graph admits a linked, componental, rooted tree-cut decomposition of finite adhesion that displays all undominated edge-ends. As a first application, we deduce that this tree-cut decomposition also displays the edge-degrees of all undominated edge-ends. For locally finite graphs $-$ where every end is an undominated edge-end $-$ this yields a linked tree-cut decomposition of finite adhesion into $\textit{finite}$ parts that displays all ends and their edge-degrees. As a second application, this latter tree-cut decomposition yields short, unified deductions of Thomassen's theorem on boundary-linked finite partitions, and of Bruhn and Stein's characterisation of Eulerian locally finite graphs in terms of even ends.

We study interactions between simplex faces of lattice polytopes and quadratic generation of toric ideals. We prove that, under a mild condition on edges, if the toric ideal of a lattice polytope is generated by quadratic binomials, then every clique of its 1-skeleton is the vertex set of a face. In particular, if the toric ideal of a $(0,1)$-polytope is generated by quadratic binomials, then every clique of its 1-skeleton is the vertex set of a face. For $(0,1)$-polytopes satisfying condition (E), we characterize this clique-face property in terms of divisibility by quadratic monomials appearing in quadratic binomials of the toric ideal; as a consequence, such toric ideals have no indispensable monomials of degree $\ge 3$. We apply these results to edge polytopes and cut polytopes, for which the clique-face property is equivalent to quadratic generation. Finally, motivated by conjectures on quadratic toric ideals, we verify the clique-face property for simple polytopes, matroid independence polytopes, and matroid base polytopes, and discuss stable set polytopes.

We prove that every $K_5$-free $n$-vertex graph with sublinear independence number can be made bipartite by removing at most $n^2(1/18+o(1))$ edges, where the constant $1/18$ is best possible. The proof method is related to extensions of Turán Theorem in edge-weighted settings, and part of the proof uses flag algebra.

In 1984, Frankl and Füredi asked for the maximum density of an $n$-vertex $r$-graph in which every $(r+1)$-set of vertices spans $0$ or $2$ edges. They gave a construction with asymptotic density $2^{1-r}$. We significantly improve this bound by constructing such $r$-graphs with density $Ω(r^{-3})$, thereby improving the dependence on $r$ from exponential to polynomial. We also obtain lower bounds for the more general problem in which every $(r+1)$-set spans an even number of edges from $\{0,2,\ldots,2k\}$.

$q$-Matroids, a $q$-analogue of classical matroids have attracted a lot of attention over the last decade, yet their enumeration remains largely unexplored. In this paper, we study the number of $q$-matroids, paving and sparse-paving $q$-matroids defined on a fixed ground space and with prescribed rank. We derive new lower bounds using constructions from constant-dimension codes and improve existing estimates. On the upper bound side, we develop two approaches: a combinatorial method based on controlling the number of dependent hyperplanes for paving $q$-matroids, and an entropy-based counting argument applicable to classes of $q$-matroids closed under contraction. These techniques yield explicit upper bounds on the logarithmic number of $q$-matroids with fixed rank and ground space. Finally, we analyze the asymptotic behavior of these bounds, and identify gaps between lower and upper estimates, leading to conjectures on the true asymptotic growth.

We continue recent work of Chao, Gabel, Larson, and Nasr in using cyclotomic polynomials for dice relabeling. In their work, one idea they expand on is finding pairs of dice with different number of sides which maintain the sum frequency of two normal dice. We continue this idea in this paper by studying pairs of dice where the number of sides of each is a different perfect square (which we call "square-sided" dice). We additionally provide conjectures offering ideas for future exploration.

The toughness of a graph is a crucial parameter for characterizing its structural properties. The toughness of a non-complete graph $G$ is defined as $τ(G) = \min \{ \dfrac{|S|}{c(G - S)} : S \subseteq V(G), c(G-S) > 1 \}$, where $c(G)$ denotes the number of components of $G$. We define $τ(K_n) = \infty$. A graph $G$ is said to be $τ$-tough if $|S| \ge τ\cdot c(G-S)$ for every vertex cut $S$ of $G$. Let $k \ge 3$ be an integer. For $\frac{1}{k-η}$-tough graphs with $η\in \{0, 1\}$, Liu, Fan and Shu \cite{a34} derived sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Jia and Lu \cite{a24}, for the case $\frac{1}{k-1} \leq τ(G) < \frac{1}{k-2}$, established sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Motivated by these results, in this paper, we further investigate sufficient conditions for the existence of a spanning $k$-tree when $\frac{1}{k} \leq τ(G) < \frac{1}{k-1}$. Specifically, for a connected $\frac{t}{t(k-1)+1}$-tough graph of sufficiently large order $n$ (where $t \ge 1$ is an integer), we provide sufficient conditions for the existence of a spanning $k$-tree in terms of the spectral radius and the signless Laplacian spectral radius. Furthermore, we establish a lower bound on the size (number of edges) to guarantee the existence of a spanning $k$-tree.

In this paper, we investigate the tree-independence number of graph classes that do not contain $K_{1,d}$ as an induced subgraph. Dallard et al. conjectured that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor has bounded tree-independence number. Our main contribution towards this conjecture is showing that the conjecture holds for outerstring graphs. Additionally we give linear and quadratic bounds for the tree-independence number of various $K_{1,d}$-free graph classes, sharpening previous bounds. Finally, we bound the tree-independence number of $K_{2,d}$-free graphs additionally forbidding holes of length at least $5$.

We study properties of graphs (or rather graph sequences) saying that some restricted count of subgraphs is approximatively what is expected in a random graph. It has been shown by several authors that many such properties characterize quasi-random graphs, but there are also some exceptions. We continue here the line of investigation in Janson and Sós (2013), and introduce some new versions of these properties, in order to better understand why many of these properties are quasi-random, and to understand the structure of the exceptions that are not. A new feature in the proofs is a simple decomposition of the subspace of symmetric functions in $L^2([0,1]^m)$ into subspaces that are irreducible for the action of measure-preserving transformations of $[0,1]$; this simplifies some arguments and gives structure to others.

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur positivity of symmetric functions naturally associated with set partitions, with respect to two different notions of descent. In the first case, the Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings. In the second case, the Schur functions correspond to two-rows Young diagrams, and the coefficients are partial sums of associated Bell numbers. A key ingredient of our approach in the second case is the notion of a removable singleton, defined algebraically and shown to admit an equivalent combinatorial interpretation via jeu-de-taquin rectification of skew tableaux. As an application, we establish Schur positivity for various classes of symmetric functions indexed by non-crossing partitions and partitions with a given number of parts. We provide an explicit combinatorial description of the tableaux that contribute to the Schur expansion, and we connects the obtained coefficients to some known integer sequences.

The study of intersecting families, initiated by Erdős, Ko, and Rado, is a central topic in extremal combinatorics. A classical stability result of Hilton and Milner determines the largest non-trivial intersecting family, and in subsequent works researchers developed structural stability results via the notion of diversity. In this paper, we study cross-intersecting families. We establish a structural theorem for large cross-intersecting pairs, extending Kupavskii's theorem from intersecting families to the cross-intersecting setting. Our result characterizes extremal cross-intersecting pairs in terms of their diversity parts and maximal cross-intersecting extensions. As corollaries, we obtain cross-intersecting analogues of several classical theorems, including those of Han--Kohayakawa and Huang--Peng. A key ingredient in the proof is a new shifting method, called the $S_{U,V}^{Q}$-shift, which not only preserves global intersection properties but also maintains certain local substructures after shifting. We expect this method to be useful elsewhere, and it is already one of the key tools in establishing a product analogue of the Hilton--Milner theorem.

For a family $\mathcal{F}$, it is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in\mathcal{F}$. We use $\mathcal{SD}(\mathcal{F}) = \{F \triangle G : F, G \in \mathcal{F}\}$ to denote the family of symmetric differences of $\mathcal{F}$. In 2023, Frankl, Kiselev and Kupavskii conjectured that for any intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n > 10k$, the inequality $|\mathcal{SD}(\mathcal{F})| \le \sum_{\ell=0}^{k-1} \binom{n-1}{2\ell}$ holds. They further observed that a proof for the range $n>3k^2$ could likely be obtained via arguments similar to those in their earlier work, though no detailed derivation was given. In this paper, we establish the conjecture under the conditions $n\ge 60k^{3/2}$ and $k\ge 50$. We also determine the extremal families, which are precisely a certain class of stars. A concentration inequality plays a central role in the proof.

In the regular three-eigenvalue setting, spectral complexity and coherent-algebraic complexity coincide: a connected regular graph has exactly three distinct eigenvalues if and only if it is strongly regular, its coherent rank is three. Although examples of regular graphs with four distinct eigenvalues and coherent rank larger than four are known, it was unknown whether coherent rank is uniformly bounded among regular graphs with four distinct eigenvalues. We show that no such bound exists, even under the additional assumption of co-edge-regularity. For every prime power \(q\), we construct infinitely many co-edge-regular graphs with exactly four distinct eigenvalues, smallest eigenvalue \(-2q-1\), and coherent rank at least \(q+4\). Consequently, coherent rank is unbounded among co-edge-regular graphs with exactly four distinct eigenvalues.

Keevash and Michaeli recently proved that, under the robustness assumption that \(G\) is an \(s\)-connector (i.e. \(\overline G\) is \(K_{s,s}\)-free), \(G\) has essentially the same multicolour Ramsey matching properties as complete graphs, with an additive error \(O(qs)\), where \(q\) is the number of colours. They asked whether the dependence on \(q\) can be removed. We answer this question in a stronger exact form. For \({\bf t}=(t_1,\ldots,t_q)\in\mathbb N_+^q\), let \(R_s({\bf t})\) be the smallest integer \(N\) such that every \(N\)-vertex \(s\)-connector \(G\) satisfies \( G\to (t_1K_2,\ldots,t_qK_2). \) We determine the exact value \[ R_s({\bf t})=\sum_{j\in[q]}(t_j-1)+ \max\left\{2s,\ s+\max_{j\in[q]}t_j\right\}. \] While Keevash and Michaeli's proof uses a compression algorithm based on the Gallai--Edmonds decomposition to reduce the colouring to a structured form, our proof is a direct minimal-counterexample argument together with a new counting method for monochromatic matchings which can be applied to \(s\)-connectors.

The clique adjacency polynomial (CAP), introduced by Soicher (2015), provides a powerful method for bounding the clique numbers of edge-regular graphs. In this paper, we extend the CAP framework to arbitrary finite graphs by expressing the relevant parameters in terms of average vertex degree and average edge-degree over potential cliques. This leads to a generalised CAP bound and an associated clique existence polynomial (CEP), which removes the dependence on an auxiliary integer variable and facilitates computation. We compare the resulting bounds with classical spectral and linear programming bounds, including those of Delsarte, Hoffman, and Haemers. We show that the generalised CAP improves upon these bounds for several families of graphs. In particular, we identify infinite families of edge-regular graphs arising from projective geometry for which the CAP outperforms the Delsarte bound, as well as families of regular and non-regular graphs where the generalised CAP improves upon the Hoffman and Haemers bounds. We also develop techniques for bounding feasible parameter regions, enabling practical application of the method to both structured and unstructured graphs.

Introduced by Ballantine, Beck, and Merca, the elementary symmetric partition function $\mathrm{pre}_k$, defined on the set of partitions with at least $k$ parts, has been a topic of recent interest. We prove that $\mathrm{pre}_k$ is injective on the set of $m$-ary partitions for positive integers $m \ge k$, generalizing the binary $k = 2$ result of Ballantine, Beck, and Merca, and complementing a result of Hadelyn, Niergarth, Li and Li showing that, for each $k \ge 3$, $\mathrm{pre}_k$ is not injective on partitions of $n$ with length $2k$ for infinitely many $n$. We introduce the skew Schur partition function $\mathrm{prs}_{λ'/μ'}$, prove injectivity results for particular choices of $λ',μ'$, and describe an application to representation theory.

The Helmholtzian of a graph $G$ is the Hodge $1$-Laplacian $L_1=L_1^{\mathrm{up}}+L_1^{\mathrm{down}}$ of its clique complex, built from the triangle--edge and edge--vertex boundary operators $\partial_2$ and $\partial_1$. Problem~5.5 of Lu, Shi, Stanić, Wang and Wang asks whether $λ_{\max}(L_1)=μ_1(G)$ for every graph $G$, where $μ_1(G)$ is the largest Laplacian eigenvalue; by the Hodge decomposition this is equivalent to $λ_{\max}(L_1^{\mathrm{up}})\leμ_1(G)$. We recast it as a question about the complement of $G$: localizing $L_1^{\mathrm{up}}$ on the cycle space of $K_n$ turns it into the inequality $λ_{\min}(\bar L|_{Z_1})\ge a(\overline{G})$, where $\bar L$ is the up Laplacian of the missing triangles of $G$ and $a(\overline{G})=n-μ_1(G)$ is the algebraic connectivity of the complement. From this viewpoint, we prove the unconditional bound \[ λ_{\max}\!\big(L_1^{\mathrm{up}}(G)\big)\ \le\ μ_1(G)+\frac13\big(n-μ_1(G)\big), \] which refines the integrality ceiling $λ_{\max}(L_1^{\mathrm{up}})\le n$ of Duval and Reiner and is sharp exactly when that ceiling is attained. We then isolate the single sharp inequality, on the dense part of $\overline{G}$, that stops the method short of Problem~5.5, and we show that the localization, the bound, and this obstruction all persist for the up Laplacian of an arbitrary finite simplicial complex, in every dimension.

For a family of graphs $\mathcal F$, a graph $G$ is said to be $\mathcal F$-free if it contains no member of $\mathcal F$ as a subgraph. A wheel graph $W_k$ is a graph on $k+1$ vertices formed by joining a new vertex to all vertices of a $k$-cycle. Given an integer $k\ge 3$, we consider the problem of determining the maximum number of triangles in a $W_{\geq k}$-free graph, where $W_{\geq k}=\{W_\ell: \ell \geq k\}$. The case $k=3$ was raised by Gallai, who proposed a conjecture for this case (see Erdős [5]. Gallai's conjecture was disproved by Zhou [17] and independently by Füredi, Goemans, and Kleitman [9]. In this paper, we study the case $k=4$. Namely, for every integer $n\ge 3$, we determine the maximum number of triangles in an $n$-vertex $W_{\geq 4}$-free graph and characterize all extremal graphs.

Let $\cpsi_{2,a}(n)$ denote the number of $(2,a)$-colored Frobenius partitions of weight $n$, where the two rows have prescribed length difference. We study the two cases $a=0$ and $a=1$ and connect them with minimal-excludant statistics on bipartitions. Let $σ\mex_2(n)$ be the sum of the Lin--Liu bipartition minimal excludants over all bipartitions of $n$, and let $E_2(n)$ be the number of bipartitions whose two component minimal excludants are equal. For all $n\geq 0$, we give a combinatorial proof of \[ \cpsi_{2,0}(n)=2σ\mex_2(n) \qquad\text{and}\qquad \cpsi_{2,1}(n)=2σ\mex_2(n)-E_2(n). \] These identities give direct combinatorial interpretations of two-colored Frobenius partition functions in terms of bipartition minimal-excludant sums.

In this article, for a convex lattice polytope, we further investigate the summatory function of its Ehrhart polynomial, which is called the summatory Ehrhart polynomial, and introduce its summatory Ehrhart series. We prove several fundamental properties of these invariants. In particular, we derive a summatory analogue of the classical Ehrhart--Macdonald reciprocity law, which establishes a signed functional equation between the polytope and its relative interior via the substitution $t \mapsto 1-t.$

We reduce the embracing exchange distance of bases of oriented matroids to the metric of oriented matroid polyhedra. This allows us to disprove recent conjectures of Caoduro, Khodamoradi, Paat, and Shepherd and of Bérczi and Nádor. On the other hand, we show that any two embracing bases of an oriented matroid of rank $r$ can be transformed into each other in at most $2r^{\log_2(r)+3}$ steps and in at most $r$ steps in a Lawrence oriented matroid, thus confirming the conjecture in this case.

In this paper we study cycles in multiset permutations and parking functions. As combinatorial objects, multiset permutations are essential building blocks for mappings and permutations, while parking functions lie between mappings and permutations. We take both algebraic and analytic views in our investigation and present exact as well as asymptotic results. We point to a surprising correspondence between two statistics on multiset permutations, terminal closers and cyclic points, shedding light on the combinatorial structure.

Median decompositions were introduced by Stavropoulos in 2015 as a generalisation of tree decompositions. In this paper, we further develop and exposit this theory as a tool in structural graph theory to study systems of vertex separations. Generalising the well-known fact that nested systems of vertex separations produce tree decompositions of a graph over the structure tree, we describe how a (not necessarily nested) system of separations produces a median decomposition. The median graph in this decomposition is the `dual median graph' constructed by Sageev. If the system of cuts is nested then this median decomposition recovers precisely the aforementioned tree decomposition. We prove a theorem asserting that this decomposition is `uniquely minimal', and describe how Sageev--Roller duality manifests in median decompositions. As an application of our structural approach, we extend a theorem of Stavropoulos from finite graphs to all graphs, which states that the median-width a graph is equal to its clique number. We also describe the link between (canonical) median decompositions and (equivariant) coarse embeddings/quasi-isometries into median graphs. A corollary of these results is a characterisation of when a finitely generated group acts metrically-properly/geometrically on a median graph, in terms of canonical median decompositions of its Cayley graphs.

One way to construct Difference Distance Magic Oriented Graphs (DDMOGs) is via a recently introduced technique called weighted sum. We explore the quality of said construction further by introducing the notion of an ornament. An ornament is an oriented graph that, when used in a weighted sum with an existing DDMOG, creates a new DDMOG. We provide results on the construction of a specific type of ornaments, called s-nodes ornaments. We conclude the paper with a list of open questions related to DDMOGs and ornaments.

The saturation number $μ^*(G)$ of a graph $G$ is the minimum cardinality of a maximal matching, and $H(G)$ is its harmonic index. TxGraffiti conjectured in 2023 that $μ^*(G) \le H(G)$ for every nontrivial connected graph $G$, and Bıyıkoğlu refuted this by showing that the ratio $μ^*(G)/H(G)$ can be made arbitrarily large. Restricting to trees bounds the ratio sharply. Every nontrivial tree $T$ satisfies $μ^*(T) < \frac{3}{2} H(T)$, with the constant $3/2$ best possible. A complementary bound $H(G) < 4μ^*(G)$ holds for every graph with an edge, so on a nontrivial tree the saturation number is pinned to $\frac{1}{4} H(T) < μ^*(T) < \frac{3}{2} H(T)$, both constants best possible. The friendship graph $F_4$ is a smallest counterexample to the conjecture, on nine vertices, and the smallest tree counterexample is the subdivided star on eleven vertices. For each positive integer $m$ a family of graphs with $m$ hubs has ratio approaching $m+1$, while the conjecture holds whenever all vertices have equal degree. Both invariants arise in applications, the harmonic index as a molecular descriptor and the saturation number as a measure of adsorption inefficiency, and the bounds estimate the latter, which is NP-hard to compute, by the former, which is computable in linear time.

We give an exact finite-dimensional Perron--Frobenius realization of the dominant zero of the Nekrasov--Okounkov polynomials $\nop _n(z)$. For a normalized positive sequence $h=(h(n))_{n\ge 1}$ with $h(1)=1$, define $\pol _0^h(z)=1$ and, for $n\ge 1$, \[ \pol _n^h(z)=\frac{z}{h(n)}\sum_{k=1}^n σ(k)\pol _{n-k}^h(z),\] where $σ(k)$ denotes the sum of divisors of $k$. The Nekrasov--Okounkov polynomials are obtained from the specialization $h(n)=n$ by the shift $\nop _n(z)=\pol _n^h(z+1)$. We derive a Hessenberg determinant representation for $\pol _n^h(z)$. After separating the trivial zero at the origin, the remaining zeros of $\pol _n^h(-z)$ are identified with the eigenvalues of an explicit $(n-1)\times(n-1)$ nonnegative matrix $M_n^h$. We prove that $M_n^h$ is primitive and apply Perron--Frobenius theory to show that $\pol _n^h(z)$ has a unique zero of maximal modulus; this zero is real, negative, and simple. As a consequence, the same property holds for the Nekrasov--Okounkov polynomials. We also prove strict monotonicity of the associated spectral radii.

We show that for any $A\subseteq \mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\subseteq \mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \[|B|^{1/m}>|A|^{1/n}.\] In particular, this implies that no fixed cap set in $\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone.

In this paper we introduce "graph linear notation" -- a complete graph invariant -- which is positioned as an alternative definition for the finite graphs. This invariant is constructed using an algorithm similar to the algorithm of finding canonical forms of graphs. Storing graph linear notation instead of a regular graph allows us to greatly simplify two major problems: the construction of illustrations for graphs with regards to possible graph symmetries, and the comparison of two graphs for isomorphism. We also demonstrate the transferability to the graph linear notations such classical graph theory concepts as colourings and graph paths.

Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \[ \{[r],[s]\}\longmapsto \{[2r],[2s]\}. \] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\equiv\pm1\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.