Papers

Gross, Mansour and Tucker introduced the partial-dual polynomial of a ribbon graph and asked under what conditions such a polynomial is even-interpolating, odd-interpolating, or both. In this paper, we provide an answer to this open problem.Using the framework of delta-matroids, we prove that the twist polynomial of any binary delta-matroid is either an even polynomial, an odd polynomial, or both even-interpolating and odd-interpolating. Applying this to ribbon graphs, we deduce that the partial-dual polynomial of any ribbon graph satisfies the same conclusion.

For a graph $G$ and an integer $d\geq 0$, the defective chromatic polynomial $\chi_d(G;k)$ counts the $k$-colorings of $G$ in which each vertex has at most $d$ neighbors of its own color. We investigate which structural properties of $G$ are determined by the full family $\{\chi_d(G;k)\}_{d\geq 0}$. We establish a contraction formula expressing $\chi_d(G;k)$ as a sum of ordinary chromatic polynomials of the edge contractions of $G$. As a first application, we prove that for triangle-free graphs, the full family determines the degree sequence. For trees, we show further that the family $\{\chi_d(T;k)\}_{d\geq 0}$ determines the path-subgraph counts $N(P_j,T)$ for $j=1,2,3,4$, but not for $j=5$. For each $n\geq 9$, we construct a pair of nonisomorphic trees of order $n$ that share the same defective chromatic polynomials for every $d\geq 0$.

We study fibres of the fertility map $\Phi$ from decorated rooted trees to decorated multi-index monomials. For a multi-index $\mathbf{k}$ of weight $-1$, the fibre $\mathcal F_{\mathbf{k}}=\{\,t:\Phi(t)=\xx^{\mathbf{k}}\,\}$ consists of all rooted trees with decoration–fertility profile $\mathbf{k}$. We consider its ordinary cardinality $F_{\mathbf{k}}$, its symmetry-weighted cardinality $W_{\mathbf{k}}$, and the coefficient mass $J_{\mathbf{k}}$ appearing in the tree expansion of the transposed embedding $\jmath$. We obtain an explicit formula and a functional equation for the weighted counts, and an exact multiset recursion together with a cycle-index functional equation for the ordinary counts. We also introduce coefficient generating functions for the lowering derivation $\bar\partial$, derive recursive and transport-array formulas for the corresponding coefficients, and use them to refine the admissible-cut formula for the coproduct in the LOT Hopf algebra.

The vertex Folkman number $F_v(s,t;k)$ is the smallest $n$ for which there exists a $K_k$-free graph on $n$ vertices whose vertices cannot be $2$-colored without producing a monochromatic copy of $K_s$ or $K_t$. We show $F_v(3,3;5)=8$. The witness is the cone $K_1 \vee \overline{C_7}$, a single universal vertex joined to the complement of a $7$-cycle. That this graph is $K_5$-free and arrows $(3,3)^v$ follows from a short independence-number argument. The matching lower bound – no $K_5$-free graph on $7$ or fewer vertices works – comes from exhaustive enumeration via nauty and a SAT check using Glucose\,4. The appendix has a self-contained Python script for verification.

Let $G$ be a simple graph with order $n$, maximum degree $\Delta(G)$, and chromatic index $\chi'(G)$, respectively. A graph $G$ is edge-chromatic critical if $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. Assume that $G$ is an $n$-vertex connected regular Class $1$ graph, and let $G^*$ be obtained from $G$ by splitting one vertex into two vertices. Hilton and Zhao in 1997 proposed the vertex-splitting conjecture: if $\Delta(G)>\frac{n}{3}$, then $G^*$ is edge-chromatic critical. Recently, Cao, Chen, and Shan (Discrete Math. 2022) verified the conjecture for $\Delta(G)\ge\frac{3n}{4}$. In this paper, we confirm the conjecture for $\Delta(G) \ge\frac{2n-2}{3}$.

It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of this inverse theorem in two regimes. First, for any parameter $M \leq n-4$, we precisely characterize the $n$-element sets of positive reals with at most $\binom{n+1}{2}+1+M$ subset sums. Second, for any constant $C$, we provide a characterization, sharp up to constants, of the $n$-element sets of positive reals with at most $Cn^2$ distinct subset sums. Along the way, we constrain (for any fixed $d \geq 2$) the structure of $n$-element subsets of $\mathbb{R}^d$ with $o(n^{d+1})$ subset sums.

In this paper, we investigate the spectrum of the unit-graph of the ring of $3 \times 3$ matrices over a finite field $\mathbb{F}_q$, which is equivalently the Cayley digraph $ \mathrm{Cay}\!\left((\mathrm{Mat}_3(\mathbb{F}_q),+), \mathrm{GL}_3(\mathbb{F}_q)\right)$. This unit-graph has a vertex set $\mathrm{Mat}_3(\mathbb{F}_q)$ with a directed edge from $A$ to $B$ whenever $B - A \in \mathrm{GL}_3(\mathbb{F}_q)$. Then, two vertices are adjacent precisely when their difference is invertible. With relevant character theory, we consequently demonstrate that the adjacency spectrum of $ \mathrm{Cay}\!\left((\mathrm{Mat}_3(\mathbb{F}_q),+), \mathrm{GL}_3(\mathbb{F}_q)\right) $ consists of four distinct eigenvalues together with their multiplicities. Using the Spectral Gap Theorem for Cayley digraphs, we show that if two subsets of vertices in $\mathrm{Mat}_3(\mathbb{F}_q)$ are sufficiently large, then there are matrices in the two subsets whose difference lies in $\mathrm{GL}_3(\mathbb{F}_q)$. In particular, any sufficiently large subset of $\mathrm{Mat}_3(\mathbb{F}_q)$ contains two distinct matrices whose difference has nonzero determinant. This spectral gap implies that large vertex sets cannot avoid each other and must be connected by at least one edge.

We introduce an affine version of Tic-Tac-Toe played on the finite affine space $\mathbb{F}_q^m$. Two players alternately claim points, and the first player to occupy all points of an affine subspace of dimension $n$ wins. We call this the $(m,n)_q$-game. For fixed $n$ and $q$, we study how the outcome depends on the ambient dimension $m$. Using strategy stealing and a blocking-set interpretation, we show that every $(m,n)_q$-game is either a first-player win or a draw, and that the property of being a first-player win is monotone in $m$. This yields a threshold $T(n,q)$: the game is a draw for $m<T(n,q)$ and a first-player win for $m\ge T(n,q)$. We prove that this threshold is finite by applying the affine/vector-space Ramsey theorem of Graham, Leeb and Rothschild, and we obtain general lower bounds from the Erdős-Selfridge criterion for Maker-Breaker games. In the binary case, we give a direct Fourier-analytic argument, combined with an inductive lifting method, which shows that \[ T(n,2)\le 2^{n+1}. \] We also determine several small cases, including $T(1,q)=2$ for $q\in\{2,3,4\}$ and $T(2,2)=4$, and we prove geometric lower bounds from explicit pairing strategies, such as $T(n,q)\ge n+2$ for every $n\ge 2$. Our results place affine Tic-Tac-Toe at the interface of strong positional games, finite geometry and Ramsey theory for finite affine spaces.

We show that every $K_4$-free graph on $n$ vertices can be made balanced bipartite by removing at most $\frac{n^2}{9}$ edges. This proves a conjecture of Balogh, Clemen, and Lidický, and generalizes both Sudakov's result on the bipartite distance of $K_4$-free graphs and Reiher's result on the sparse half of $K_4$-free graphs.

Suppose one has a party of $m$ people, whose expertise collectively covers $n$ topics. Given a subset $T$ of the topics, one wishes to form a panel of $|T|$ people from the party such that $T$ can be covered by assigning a distinct topic to each panel member with the expertise. We show that the numbers of such panels, as $T$ varies, form a Lorentzian polynomial. We achieve this by showing that a certain linear operator on polynomials, which we call the “inducing operator” for its connection to induced (poly)matroids, preserves Lorentzian polynomials and realizable volume polynomials.

Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have \[ \Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} \alpha_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p}, \] where $c=2$, $1/p+1/p'=1$, and $\alpha_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$. In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$. We then prove that if an estimate of the above form holds, the exponent must satisfy $c\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\ge 2$. In the second part of the paper we obtain a sharp three-function bound \[ \Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2\Gamma^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p}, \] where $p \geq 3$, $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $\Gamma=\Gamma(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.

Let $(\Gamma,+)$ be an Abelian group of order $n^2$. A $\Gamma$-magic square of order $n$ is an $n\times n$ array whose entries are pairwise distinct elements of $\Gamma$ such that all row sums, column sums, and the two main diagonal sums are equal to the same element $\mu \in \Gamma$, called the magic constant. A combinatorial design is called $\Gamma$-additive if its point set is a subset of an Abelian group $\Gamma$ and every block has sum zero. If the point set coincides with $\Gamma$, the design is said to be strictly $\Gamma$-additive. Motivated by this notion, we construct $\Gamma$-magic squares with magic constant $\mu=0$ whose rows, columns, and two main diagonals can be used as blocks of a strictly $\Gamma$-additive design. We call such a square zero-sum $\Gamma$-magic square. In this paper, we establish necessary and sufficient conditions for the existence of zero-sum $\Gamma$-magic squares.

We present new degree-sequence lower bounds on the expected size of an independent set from the hard-core model. For arbitrary graphs, we establish a multivariate lower bound inspired by a conjecture of the first author and Kang and a recent bound on the multivariate partition function due to Lee and the third author. By applying a novel spectral analysis to the local occupancy linear program, our method successfully bypasses the convergence radius limitations of the cluster expansion and avoids induction. For graphs with bounded local maximum average degree, including triangle-free graphs, we prove a univariate bound extending prior work by a subset of the authors. In both cases our bounds require the fugacities to be upper bounded by $c/\Delta$ where $\Delta$ is the maximum degree of the graph and $c$ is an absolute constant depending on the setting.

Let $G$ be a finite abelian group of order $n$ and let $\mathcal M_G=(x_{a+b})_{a,b\in G}$ be the Cayley table of $G$. Let $\text{imm}_\lambda(\mathcal M_G)$ be the immanant of $\mathcal M_G$ with respect to a partition $\lambda$ and $\mathcal I_\lambda(G)$ be the number of formally different monomials occurring in $\text{imm}_\lambda(\mathcal M_G)$ (in particular, we denote by $\mathcal P(G)$ (resp. $\mathcal D(G)$) for the corresponding quantity for $\text{per}(\mathcal M_G)$ (resp. $\text{det}(\mathcal M_G)$) for simplicity). The study of $\mathcal P(G)$ and $\mathcal D(G)$ lies at the intersection of algebraic combinatorics and additive combinatorics. In this paper, we prove the following results. (1) If $|G|$ is a prime power, then $$\mathcal P(G)=\mathcal D(G).$$ (2) If $|G|$ is odd, then $$\mathcal I_{(n-1,1)}(G)= \mathcal I_{(2,1^{n-2})}(G)=0,$$ and if $|G|\equiv 2\pmod 4$, then $$\mathcal I_{(n-1,1)}(G)=\mathcal P(G)\quad \text{and}\quad \mathcal I_{(2,1^{n-2})}(G)=\mathcal D(G).$$ (3) If $|G|$ is odd and $|G|\ge 7$, then $$ \text{imm}_{(4,1^{n-4})}(\mathcal M_G)=\text{imm}_{(2,2,2,1^{n-6})}(\mathcal M_G).$$

We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of their quotient admits a simple closed form. This allows us to prove several conjectures of McIntosh on asymptotic transformations of $q$-series. Moreover, our results extend some identities of Ramanujan and McIntosh.

The purpose of this paper is twofold. First, we introduce a family of generalized Markov-Hurwitz equations, extending classical Markov-Hurwitz equations with additional degree n-1 interaction terms, Gyoda and Matsushita's generalized Markov equations from 3 variables to n variables. Second, we prove a logarithmic asymptotic phenomenon for the positive integer solutions of these equations.

Let $G$ be an $n$-vertex graph, and let $\lambda(G)$ and $\lambda_n(G)$ denote the largest and smallest eigenvalues of its adjacency matrix. Write $e(G)$ for the number of edges of $G$, $d(G)=2e(G)/n$ for its average degree, and $T_r(n)$ for the $r$-partite Turán graph on $n$ vertices. We prove four sharp results in spectral Turán theory. First, we confirm Guiduli's spectral dense-neighborhood conjecture (1996) in a stronger form: if $\lambda(G)\ge \lambda(T_r(n))$, then either $G\cong T_r(n)$, or there exists a vertex $v$ such that $\lambda(G[N(v)]) > \lambda(T_{r-1}(d(v)))$. Moreover, when $\lambda(G)>\lambda(T_r(n))$, every vertex attaining the maximum entry in any nonnegative Perron eigenvector of $G$ has this property. Second, we answer a problem of Nikiforov (2009) by showing that the exact Turán edge threshold is detected by the exact spectral threshold: for every $r\ge 2$ and every $n$, $\lambda(G)<\lambda(T_r(n))$, implying $e(G)<e(T_r(n)).$ Our proof also determines the equality cases. Third, we answer another question of Nikiforov (2009) by showing that his least-eigenvalue clique bound \[ \omega(G)\ge 1+\frac{2e(G)}{(n-d(G))(d(G)-\lambda_n(G))} \] does imply the concise form of Turán's theorem. Finally, we discuss an open problem proposed by Ai et al. (2026) in \cite{ALNS26+}.

We propose a local transformation on bicolored graphs, which we call local homophily, inspired by adaptive networks and based on majority dynamics and homophily. In this transformation, a vertex updates its color to match the majority of its neighbors, while neighbors of the same color become connected and neighbors of the opposite color become disconnected. We show how to simulate Boolean circuits using local homophily and establish that determining whether a given pair of vertices becomes connected under iterative applications of local homophily is $\mathbf{P}$-complete under logspace reductions.

We say that two permutations $[n]\to [n]$ intersect if they map some element $x$ to the same element $y$. A matching in a family of permutations is a collection of pairwise disjoint permutations. In this paper, we study families of permutations with no matchings of size $s$. In particular, we obtain a characterization of the largest $s$-matching-free families and a Hilton–Milner type result. We also obtain results for the families of derangements.

We provide multiple combinatorial expansion formulas - in terms of snake graphs, labelled posets, matrices, and $T$-walks - for elements in generalized cluster algebras associated to arcs on punctured orbifolds and illustrate their equivalence. This work generalizes and unifies existing work on combinatorial expansion formulas from surfaces and unpunctured orbifolds.

We develop a homotopy theory of directed graphs based on cubical homotopy groups, also known as $A$-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups yields an $\infty$-category, denoted by ${\sf DGra}_\infty$. We prove that ${\sf DGra}_\infty$ is equivalent to the $\infty$-category of spaces, establishing a directed version of the discrete homotopy hypothesis of Carranza and Kapulkin.

We prove that for every $t \in \mathbb{N}$, prime-length cycles do not have the $\frac{1}{t}$-integral Erdős-Pósa property, even when restricted to planar graphs. We in fact prove a more general density result. For every $t \in \mathbb{N}$ and every subset $L \subseteq \mathbb{N}$ with lower density zero, the set of cycles whose length is in $L$ do not have the $\frac{1}{t}$-integral Erdős-Pósa property, even when restricted to planar graphs. We also consider a less restrictive density condition on $L$, called porous, where the complement of $L$ contains arbitrarily long sequences of consecutive integers. We prove that for every porous set $L \subseteq \mathbb{N}$, the set of cycles whose length is in $L$ do not have the Erdős-Pósa property, even when restricted to projective planar graphs. Our results partially answer a question of Gollin, Hendrey, Kwon, Oum, and Yoo [Math. Ann., 393(2):2507-2559, 2025].

Andrews investigated parity conditions in the Rogers-Ramanujan-Gordon theorem. Under the conditions that even parts or odd parts appear an even number of times, Andrews discovered two Rogers-Ramanujan-Gordon type partition theorems and derived corresponding generating functions. In the Rogers-Ramanujan-Gordon theorem, there are two parameters $k$ and $a$, where $k-1$ is the maximum number of consecutive parts $l$ and $l+1$, and $a-1$ is the maximum number of parts equal to $1$. Andrews' first theorem deals with the case $k\equiv a \;(\rm{mod}\;2)$, while the second theorem concerns the case where $k$ is even and $a$ is odd. These two partition identities have different infinite product forms on the right-hand side. In this paper, we consider the case $k\not\equiv a \;(\rm{mod}\;2)$ and use Bailey's lemma to obtain an Andrews-Gordon type identity whose right-hand side coincides with that of Andrews' identity for the case $k\equiv a \;(\rm{mod}\;2)$. By fixing the number of peaks of the corresponding lattice paths, we also derive a recurrence system whose solution agrees with the product-side generating function. We were unable to find a suitable combinatorial interpretation of the infinite sum form of this expression in terms of partitions, but with the help of lattice paths, we provide an appropriate combinatorial interpretation. Finally, we prove this identity analytically by applying Bailey's lemma.

A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow\{0,1,2,\ldots,k\}$ with $\sum\limits_{e\in E_G(v)}f(e)\leq k$ for each vertex $v$ of $G$, where $E_G(v)$ is the set of edges incident with $v$ in $G$. A perfect $k$-matching of a graph $G$ is a $k$-matching $f$ satisfying $\sum\limits_{e\in E_G(v)}f(e)=k$ for any vertex $v$ of $G$. A fractional perfect matching of a graph $G$ is a function $f:E(G)\rightarrow [0,1]$ satisfying $\sum\limits_{e\in E_G(v)}f(e)=1$ for any $v\in V(G)$. We denote by $\rho(G)$ the spectral radius of $G$. In this paper, we put forward a tight spectral radius condition for a $t$-connected graph to possess a perfect $k$-matching and a tight spectral radius condition for the existence of a perfect $k$-matching in a $t$-connected graph with a fractional perfect matching.

Multilinear varieties, defined as the sets of rational points of varieties cut out by multilinear functions, were first introduced and studied by Gowers and Milićević[Proc. Edinb. Math. Soc., 2021] for finite $\mathbb{K}$. In this paper, we investigate multilinear varieties over infinite fields from a geometric perspective. We establish two fundamental results: a codimension formula for the Zariski closure of a multilinear variety, and the existence of a high-dimensional irreducible subvariety passing through any given $\mathbb{K}$-rational point. These results serve as a geometric foundation for analyzing various ranks of tensors and homogeneous polynomials, including partition rank, analytic rank, geometric rank, (collective) strength and (collective) Birch rank. As applications, we resolve the Adiprasito-Kazhdan-Ziegler conjecture [arXiv:2102.03659, 2021] on the stability of partition rank for perfect infinite fields. We thereby settle the stability conjecture for collective strength [Selecta Math., 2024], as well as the conjecture on the linear equivalence between strength and Birch rank [arXiv:2410.00248, 2024] for such fields. Moreover, our results immediately yield a strengthening of the theorems of Bik-Draisma-Snowden [arXiv:2401.02067, 2024] and Lampert-Snowden [arXiv:2406.18498, 2024], for multilinear varieties over infinite fields.

We study the class of edge-coloured graphs arising from the graph-theoretic representation of quantum photonic experiments that generate multipartite W-states. Abstracting away physical amplitudes and phases, we introduce W-state graphs: matching-covered graphs equipped with a half-edge 2-colouring such that every perfect matching contains exactly one bichromatic edge and every vertex is incident with a red half-edge. Our main contribution is a complete structural characterization of W-state graphs. We show that a graph is a W-state graph if and only if each of its 3-connected components is a W-cone, a simple and rigid building block defined by a universal vertex and a factor-critical base. This characterization implies that no W-state graph is simple and yields a recognition algorithm running as fast as verifying whether a graph is matching-covered. We also show that the natural generalization to Dicke states encounters a complexity barrier: verifying one of the two Dicke state conditions is itself coNP-complete, resolving an open problem of Vardi and Zhang [IJCAI 2023]. Our results place W-state graphs firmly within classical matching theory and precisely delineate the combinatorial structures capable of realizing idealized W-states in the experiment-graph framework.

This paper investigates the energy and vertex energy of the modified divisor prime graph $G^*_{Dp}(n)$, which is distinguished from the standard divisor prime graph by the inclusion of a self-loop at the vertex $1$. To facilitate this analysis, we introduce a generalized definition of vertex energy for graphs with self-loops and demonstrate its mathematical consistency.

We show that the class of all finite regular tournaments is cofinal in the class of finite tournaments. In addition, we establish cofinality results for certain special subclasses of regular tournaments. We also provide an algorithm for constructing these regular tournaments.

We consider an Erdős-Ko-Rado type sum that weights each member of a uniform family according to its smallest intersection with the rest of the family. We prove that once the ground set is sufficiently large this sum is at most one, with equality exactly for stars. This simultaneously generalizes the usual Erdős-Ko-Rado theorem for every intersection threshold $t$ and $n$ sufficiently large. As a consequence we also obtain an extension of Hilton's theorem on cross-intersecting families.

Given graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs. The friendship graph $F_k$ consists of $k$ triangles sharing a common vertex. In this paper, we determine the value of $\mathrm{ex}(n,K_3,(t+1)F_k)$, where $K_3$ is a triangle, $t\geq 1$ is an integer, and $(t+1)F_k$ denotes a union of $(t+1)$ pairwise vertex-disjoint copies of $F_k$. Moreover, we characterize the extremal structure. Our result can be viewed as a generalization of the result of Zhu, Chen, Gerbner, Győri, and Hama Karim, as well as of the remaining case left open by Wang, Ni, Liu, and Kang. In contrast to the extremal graphs of $F_k$, the extremal graphs of $(t+1)F_k$ undergo a fundamental change. This structure is also different from those of previous similar problems.