Papers

We prove the antipode formula for the $q$-deformations of quasi-symmetric functions. We also define a fundamental basis for non-commutative quasi-symmetric functions, and establish a partial antipode formula there. Our method also reduce to a new proof for the known antipode formula on the usual quasi-symmetric functions.

We give an exact solution of the ferromagnetic Ising model on a random regular graph ensemble via analytic combinatorics. Expressing the partition function as the generating function of labeled edge-bicolored graphs, we obtain the free energy in the thermodynamic limit from the asymptotic enumeration of these graphs. A simple analysis of the resulting formula reveals a second-order phase transition with critical exponents of the mean-field universality class.

Given a graph $G$ and an integer $\ell>\chi(G)$, the reconfiguration graph of the $\ell$-colorings of $G$ has as its vertices as the proper $\ell$-colorings of $G$, with an edge between two colorings whenever they differ on exactly one vertex. We say that $G$ is recolorable if this reconfiguration graph is connected for every $\ell>\chi(G)$. Belavadi and Cameron determined which $(F_1,F_2)$-free graphs are recolorable whenever $F_1$ and $F_2$ are graphs on at most four vertices, with the single exception of $(2K_2,K_4)$-free graphs. Gaspers and Huang showed such graphs are $4$-colorable. The $3$-colorable case within this class has also been resolved, leaving the open question of whether every $(2K_2,K_4)$-free graph with chromatic number $4$ is recolorable. In this paper, we provide evidence toward an affirmative answer by establishing recolorability for three subclasses: $(2K_2,K_4,C_5)$-free graphs, $(2K_2,K_4,H_a,H_b)$-free graphs for any distinct $a,b\in \{2,3,4\}$, and $(2K_2,K_4,H_4)$-free graphs containing an induced $W_5$, where $H_i$ denotes the unique $2K_2$-free graph obtained from a $W_5$ by keeping exactly $i$ edges from the universal vertex to the cycle.

We prove two inequalities introduced in our prior study of the graphical shape of the number of bit strings with a given score under an interesting scoring system. Generating functions are used to establish the inequalities, which in turn imply two of the salient graphical features, uni-modality near the zero score and shape asymmetry for positive versus negative scores. One inequality provides a lower bound on the expected value of a discrete random variable with probabilities proportional to a product of two binomial coefficients. The other inequality states that the expected value with respect to near central binomial coefficients of other binomial coefficients lying on an oblique ray in Pascal's triangle exceeds the expected value along an adjacent parallel ray to its left.

We show that the separation profile of a hyperbolic planar graph and, more generally, a hyperbolic apex-minor-free graph, grows at most logarithmically, answering a question of Benjamini, Schramm, and Timár in the affirmative.

We approach discrete and differential Faà di Bruno formulas from a Möbius inversion angle. On the Boolean cube, Newton's discrete Taylor formula and the definition of iterated forward differences form a zeta–Möbius dual pair, and composing two Taylor expansions and inverting once yields a closed discrete Faà di Bruno formula at a fixed basepoint: for arbitrary maps $f, g$ between abelian groups, $$ \Delta(f \circ g;\,x;\,u_1,\dots,u_k) = \sum_{H \in \mathrm{Cov}(k)} \Delta(f;\,g(x);\,(\Delta(g;x;u_T))_{T\in H}), $$ where $\mathrm{Cov}(k)$ denotes the coverings of $[k]$ by nonempty subsets. Grouping repeated directions gives binomial versions on multi-index grids, and iterating gives formulas for $m$-fold composites, with integer covering coefficients governed by explicit cross and level recursions, a discrete analogue of the Constantine–Savits formulas. The relationship between coverings and partitions appearing in classical Faà di Bruno formulas is exhibited in an algebraic setting. The discrete formulas are Taylor expansions over the function algebra of the Boolean cube, whose idempotent generators absorb overlapping products; in the differential analogue nilpotent generators annihilate overlaps and only partitions remain. We demonstrate how these algebraic identities can be lifted to the analytical setting of $C^n$ maps between Banach spaces, recovering the multivariate Faà di Bruno formula of Constantine–Savits and extending it to composites of several maps. Boolean finite differences, binomial grid formulas, infinitesimal Taylor algebras, and Fréchet derivatives thus appear as four realizations of one Möbius-dual Faà di Bruno formula, connected by a flat family.

The induced Erdős–Pósa property in graphs relates the maximum number of pairwise anti-adjacent copies of an object with the minimum number of neighborhoods required to hit all copies. In this paper, the objects we consider are long cycles and long thetas, both as induced minors. Let $C_t$ denote the cycle with $t$ vertices and let $\Theta_t$ be the graph consisting of three internally disjoint and anti-adjacent paths, each with $t$ internal vertices, connecting the same pair of distinct vertices. We show that for every fixed $t$, both $C_t$ and $\Theta_t$ have the induced Erdős–Pósa property with respect to the induced minor relation. More precisely, for every integer $k$ and every graph $G$, one of the following two outcomes occurs: (i) $G$ contains $k$ pairwise vertex-disjoint and anti-adjacent copies of $C_t$ (resp., $\Theta_t$) as induced minors, or (ii) there is a set $X \subseteq V(G)$ of size $\mathcal{O}(tk \log k)$ such that the set $N[X]$, consisting of $X$ and its neighbors, hits all $C_t$ (resp., all $\Theta_t$) induced minors in $G$. This resolves in a strong form a special case of a conjecture of Ahn, Gollin, Huynh, and Kwon [SODA 2025]. From these results we derive that graphs that exclude $k$ disjoint copies of $\Theta_t$ as an induced minor admit balanced separators consisting of the neighborhood of $\mathcal{O}(tk \log k)$ vertices. This in turn resolves a special case of a conjecture of Gartland and Lokshtanov and, combined with known techniques, yields a QPTAS for Maximum Weight Independent Set and a number of its generalizations.

How close to singularity can an $n \times n$ unimodular matrix be? For ternary cases as $n$ increases, exact expressions are unlikely, but upon fixing $n=4$ and assessing $(2k+1)$-ary cases as $k$ increases, we make significant progress; similarly for $(k+1)$-ary cases of $4\times 4$ nonnegative unimodular matrices.

Schubert calculus on complex Grassmannians can be played by means of differential operators acting on Schur polynomials or Vertex Operators acting on exterior algebras. In this paper we develop this point of view systematically and complement it with a parallel exterior-algebra formalism, leading to what we call, respectively, the bosonic and the fermionic Schubert calculus. The two alluded realizations are related by (a finite type version of) the boson–fermion correspondence, thereby providing a unified framework connecting Schubert calculus and integrals on the Grassmannian, symmetric functions, exterior algebras and the representation theory of symmetric groups.

We describe the set of palindromic Poincaré polynomials in $B_n$. As a key ingredient in this classification, we prove that $A_n$ embeds into $B_n$ as a lower interval in Bruhat order. Consequently, all Poincare polynomials $P_w(q)$ and all Kazhdan–Lusztig polynomials $P_{x,y}(q)$ of $A_n$ occur in $B_n$.

For integers $l>0$ and $m\geqslant0$, we introduce the numbers $$S_l^{(m)}(n) = \sum_{k_1,\ldots,k_l\in\mathbb N\atop k_1+\cdots+k_l = n} \binom n{k_1,\ldots,k_l}^m \ \ (n=0,1,2,\ldots),$$ and prove that for any prime $p$ not dividing $l+1$ we have the congruence $$\sum_{n=1}^{p-1}\frac{(-1)^{mn}}{n^{m-1}}S_l^{(m)}(n)\equiv0\pmod p.$$ We also obtain a $q$-analogue of this result. For the Domb numbers given by $$D(n)=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}=S_4^{(2)}(n)\ \ (n=0,1,2,\ldots),$$ we confirm a previous conjecture which states that $$\sum_{n=1}^{p-1}\frac{D(n)}n\equiv\left(\frac p3\right)\frac 25pB_{p-2}\left(\frac13\right)\pmod{p^2}$$ for any prime $p$, where $(\frac p3)$ is the Legendre symbol, and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.

A classical notion of a factor complexity of an infinite word is defined as a function $p(n)$ counting, for each $n$, the number of distinct factors (or blocks of consecutive letters) of the word of length $n$. The notion has various generalizations and variants. For example, the abelian complexity $p_{ab}(n)$ counts the number of distinct factors of each length $n$ up to abelian equivalence, i.e., only the numbers of occurrences of letters are taken into account, and not their order. The notion of a group complexity generalizes both notions of a factor and an abelian complexities. Namely, given a sequence $\omega=(G_n)_{n=1}^{\infty}$ of subgroups of the symmetric group $S_n$, the group complexity $p_{\omega}(n)$ of a word counts the number of classes of factors of each length $n$ of the word, where words obtained from one another by permutations from $G_n$ are put in the same class. Taking $G_n=S_n$, we obtain the abelian complexity, and taking $G_n=Id$, we recover the factor complexity. Clearly, the group complexity value is between the abelian and the factor complexities. In this paper, we are interested in the following property of words. We say that an infinite word has universal group complexity if for each length $n$ and for each $k$ satisfying $p_s^{ab}(n) \leqslant k \leqslant p_s(n)$, there exists a group $G \in S_n$ such that $p_s^G(n) = k$. In other words, all “intermediate” values of complexity can be obtained. We show that Sturmian words satisfy the universal group complexity property, while they are not the only ones. We also study the universal group complexity property for aperiodic ternary words of minimal complexity and for eventually periodic words.

In this paper, we introduce a framework that aims to unify classes defined by forbidden induced subgraphs or induced minors with classes defined by the existence of certain structured tree decompositions. Let $\mathcal{H}$ be a fixed family of graphs. We define \emph{induced-$\mathcal{H}$-packing treewidth}, a tree-decomposition-based graph parameter that, for each bag, measures the maximum number of pairwise anticomplete induced copies of graphs from $\mathcal{H}$ intersecting that bag. This notion generalizes some previously studied parameters: when $\mathcal{H}=\{P_1\}$, it is equivalent to tree-independence number, and when $\mathcal{H}=\{P_2\}$, it is equivalent to induced matching treewidth. We show that bounded induced-$\mathcal{H}$-packing treewidth yields new algorithmic consequences for a range of choices of $\mathcal{H}$. In particular, we prove the following results for graphs of bounded induced-$\mathcal{H}$-packing treewidth. Our results partially answer and substantially extend a question of Bodlaender, Fomin, and Korhonen [SODA~2026] on the tractability of MWIS for graphs of bounded induced-$\mathcal{H}$-packing treewidth for $\mathcal{H}=\{P_3\}$ and for $\mathcal{H}$ equal to the family of all cycles.

We study multiplicative graph spanners in the $f$-degree fault tolerant ($f$-DFT) model, in which the spanner must approximately preserve distances even after any subset of edges of maximum degree $f$ temporarily "fails" and is removed from the graph. We prove that there are $n$-node lower bound graphs for which any $f$-DFT $(2k-1)$-stretch spanner $H$ must have size $$|E(H)| \ge \Omega\left( f^{1-1/k} n^{1+1/k}\right).$$ This matches a lower bound that was previously only known to hold conditionally, under the 1963 girth conjecture of Erdős. It also matches the current upper bounds, up to a factor of $\texttt{exp}(k)$. Our proof is an analysis of the so-called Wenger graphs (J. Comb. Theory 1991), via their recent reinterpretation by Szabó and by Conlon (Am. Math. Monthly 2021).

In a 2021 survey on Katona's circle method, Frankl conjectured that every family $\mathcal{F}\subseteq 2^{[n]}$ in which any two members intersect and no two members cover $[n]$ satisfies the sharp binomial norm bound $ \lVert\mathcal F\rVert_n :=\sum_{F\in\mathcal F}\binom{n}{|F|}^{-1} \leq \frac{n+1}{6}. $ This improves the earlier estimate $\frac{n}{4}$ obtained by the circle method. In this paper, we prove Frankl's conjecture and determine all extremal families. Our proof develops a continuous $p$-biased measure approach in place of the circle method. The intersection and union conditions lead to a sharp estimate for $ \mu_p(\mathcal F)+\mu_{1-p}(\mathcal F). $ Integrating this estimate over $p$ converts it directly into the desired binomial norm bound and recovers the optimal coefficient $\frac{1}{6}$. This continuous averaging is the key new ingredient of the proof and also yields the characterization of all extremal families.

We define an invariant of the module category of a representation-finite algebra by the magnitude of its Auslander algebra. This invariant will be called the magnitude of the module category. For bound path algebras, it can be computed as the Euler characteristic of the Auslander–Reiten quiver, in a suitable sense. To aid the computation of our invariant, we define the Auslander–Reiten–Euler characteristic of a translation quiver. We build on classical results in Auslander–Reiten theory to determine the magnitude of module categories of biserial algebras, hereditary path algebras, radical square zero bound path algebras, and self-injective bound path algebras. In these cases, we express our invariant in terms of other known quantities, notably the rank of the Grothendieck group and Coxeter numbers of Dynkin quivers. Based on our calculations and results, we obtain a conjectural characterisation of representation-finite biserial algebras in terms of the magnitude of the module category and the rank of the Grothendieck group.

We study the dimer model on two families of non-bipartite graphs on a torus. The first family is obtained by replacing degree $3$ vertices in a bipartite torus graph with triangles, while the second consists of corner graphs associated with bipartite torus graphs. We determine the relationship between the Newton polygons of these graphs and those of the underlying bipartite graphs. We also identify the primitive edge vectors of the Newton polygons with the homology classes of the zig-zag paths. We further consider the marginal polynomials obtained by restricting to monomials corresponding to a boundary side of the Newton polygon. For the triangular lattice and the Fisher graph of the hexagonal lattice, we prove that these polynomials are real-rooted and obtain an explicit factorization of their roots. Finally, we introduce new local moves, including a move on non-planar graphs, that preserve the dimer partition functions up to a scale.

We study Schubert subspace codes, which are constant-dimension subspace codes with prescribed intersection conditions with a fixed subspace. Our goal is to construct codes of maximum possible size in the extremal distance cases where a natural counting upper bound applies. We give two families of constructions. The first one uses a direct-sum decomposition of the ambient space, together with partial spreads and colorings of powers of $q$-Johnson graphs. For this construction, we also prove necessary conditions, which show how chromatic and clique obstructions arise. The second family is obtained by field reduction from evasive and scattered subspaces over extension fields. This gives codes whose size can be computed exactly in the scattered case and recovers the only previously known construction as a special case.

We give a systematic analysis of the analytic spread of the determinantal ideal $J_{G,H}$ arising from a pair of graphs $(G, H)$. We give sharp bounds for this analytic spread, and combinatorial conditions and obstructions for its maximality via a linear matroid. When $H$ is a single edge, $J_{G,H}$ is isomorphic to the binomial edge ideal $J_G$ and its analytic spread is shown to equal the rank of $G$ in Kalai's $2$-hyperconnectivity matroid.

The Graham-Pollak theorem says that one needs at least $n - 1$ complete bipartite graphs to cover each edge of a complete graph $K_{n}$ on $n$ vertices exactly once. The odd cover problem is a parity analogue which seeks the minimum number of complete bipartite graphs, denoted by $b_2(n)$, such that each edge of $ K_n $ is covered an odd number of times. An odd cover of a complte 3-graph $K_n^{(3)}$ on $n$ vertices is a family of complete $3$-partite $3$-graphs such that every triple is covered an odd number of times. Let $b_3(n)$ be the minimum size of such a family. The values of $b_2(n)$ and $b_3(n)$ are determined for some $n$ in several previous works. In this paper, we first determine the value of $b_2(n)$ for all $n$, which confirms a conjecture due to Buchanan et al. (JGT, 2026), and then show $b_3(n+1)=b_2(n)$ by which the value of $b_3(n)$ is determined for all $n$, that resolves a question posed by Leader and Tan (EJC, 2026).

In this paper, we introduce the multiset partition dimension of graphs. This parameter extends the classical partition dimension to the multiset setting by considering multiset distances from vertices to parts of a vertex partition. We establish some fundamental properties of this parameter and determine its exact values for several important classes of graphs.

Network realignment complexes were introduced by Kozlov. We generalise their definition to arbitrary connected base graphs. For a connected graph $G$, we characterise the connected components of the associated network realignment complex $X_G$ and show that $X_G$ admits an $\operatorname{Aut}(G)$-equivariant strong deformation retraction onto the disjoint union of a complete graph and a discrete $\operatorname{Aut}(G)$-space. For the complete base graph $K_n$, we study the metric structure of the network realignment graph $\mathcal{G}_n$ and obtain explicit upper and lower bounds for its diameter. Finally, we prove that $X_n$ is a cubical flag complex and that every automorphism of $X_n$ is induced by a relabelling of the underlying vertex set. In particular, $\operatorname{Aut}(X_n)\cong S_n$ for all $n\geq 5$.

The Erdős Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. In this paper, we study a spectral counterpart of this conjecture. For sufficiently large $n$, we determine the maximum spectral radius over all $n$-vertex $r$-uniform hypergraphs whose matching number is less than $s$, and characterize the unique extremal hypergraph. To establish the main theorem, we first apply the shifting method to reduce the problem to shifted hypergraphs. We then derive several spectral upper bounds through hypergraph decomposition and related variational estimates for tensor spectral radii. With these estimates, we analyze the structural properties of shifted-saturated hypergraphs and prove the spectral extremal theorem for shifted hypergraphs with bounded matching numbers. Finally, we drop the shifted condition and extend our spectral bound to general $r$-uniform hypergraphs. Our main theorem states that for any $n$-vertex $r$-uniform hypergraph $H$ with matching number $\nu(H)<s$, the inequality $\rho(H)\leq \rho(\mathcal{F}_{s-1}(n))$ holds whenever $n$ is sufficiently large. Here $\mathcal{F}_{a}(n)$ denotes the family of all $r$-subsets of $[n]$ intersecting the vertex set $[a]$, and equality is attained if and only if $H$ is isomorphic to $\mathcal{F}_{s-1}(n)$. As an immediate corollary, we derive a spectral counterpart of the classical Erdős-Ko-Rado theorem for intersecting hypergraph families.

In this paper, we prove the existence of directed strongly regular graphs with parameters (108,11,3,2,1), (108,14,10,0,2), (108,22,12,6,4), (108,23,9,8,4), (108,25,15,8,5), (108,34,18,12,10), (108,38,22,12,14), (108,39,23,14,14), (108,41,35,16,15), (108,42,33,18,15) and (108,46,22,19,20). Further, we obtain directed strongly regular graphs with parameters (54,10,4,1,2), (54,11,4,3,2) and (54,16,7,4,5). The constructions are obtained by considering finite groups acting transitively on 54 and 108 vertices.

Friedman and Wigderson (1995) introduced a notion of second eigenvalue for hypergraphs that generalizes the second eigenvalue of the adjacency matrix of a graph. We show that $r$-uniform Erdős-Rényi hypergraphs on $n$ vertices exhibit a spectral gap as soon as their expected number of hyperedges $m$ satisfies $m \gg n^{r/2}$. Prior work identified this scale only up to logarithmic factors; removing these factors is the main technical challenge. Our proof overcomes this obstacle through an explicit decomposition of an associated selector process, inspired by a generic decomposition theorem of Talagrand (2021). As a consequence of our techniques, we obtain improved norm bounds for sparse random tensors with independent entries. Finally, under a mild moment equivalence assumption, we extend to tensors a seminal result of Seginer (2000) for random matrices with i.i.d. entries.

For coprime integers $\ell,\mu\ge 2$, Alanazi, Munagi, and Saikia (2026) studied $\overline{R}_{\ell,\mu}(n)$, the number of overpartitions of $n$ in which no part is divisible by $\ell$ or by $\mu$, together with the single-modulus analogue $\overline{R}_{\ell (n)$. We record a simple combinatorial mechanism that determines both functions modulo every power of $2$ in terms of the number of distinct part sizes of the underlying ordinary partition. We also deduce a clean characterization of $\overline{R}_{\ell}(n)$ and $\overline{R}_{\ell,\mu}(n)$ modulo $4$ in terms of perfect squares.

Fried recently enumerated Dyck paths having equally many black and white cells below them, for the chessboard coloring (Narayana numbers) and the column-alternating coloring (Fuss–Catalan numbers). We prove a generalization here: for the coloring of columns modulo any $c\ge2$, the number of Dyck paths of semilength $n$ whose $c$ residue classes carry equal weight is the Raney number $\Raney_{c+1,r}(m)$, where $n=cm+r-1$.

Agraph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases its toughness, where t is a positive real number. It is conjectured that every $(K_1\cup P_4)$-free 1-tough graph is hamiltonian. In this paper, we characterize the structure of minimally 1-tough $(K_1\cup P_4)$-free graphs, and thus show that the above conjecture is true for minimally 1-tough graphs. Furthermore, it is also proved that the Kriesell's conjecture which states that each minimally 1-tough graph has a vertex of degree 2 holds for minimally 1-tough $(K_1\cup P_4)$-free graphs.

We extend the Schanuel integration framework, originally introduced for finite unions of convex sets, to arbitrary semi-algebraic sets. We prove that the resulting Schanuel integral of indicator functions is independent of the choice of ordered linear bases and therefore defines a well-defined Euler characteristic in the semi-algebraic category. We further show that this Schanuel–Euler characteristic coincides with the classical Euler characteristic defined via Borel–Moore homology and cylindrical algebraic decomposition. The recursive fiberwise structure of Schanuel integration provides an elementary and geometric interpretation of Euler characteristic and yields simplified proofs of several classical properties, including invariance under semi-algebraic isomorphisms.

Given a graph $H$, the minimum feedback vertex number of $H$ is the minimum number of vertices whose removal results in an acyclic graph. In this paper, we investigate Turán-type extremal problems for bipartite graphs in terms of their feedback vertex number. Our first result concerns bipartite graphs $H$ with minimum feedback vertex number one. Such graphs can be obtained from a forest by identifying a specified collection of leaves into a single vertex. For these graphs, we show that $\text{ex}(n, H)$ is upper bounded by $O(n^{1+1/k^\ast})$, where $2k^\ast$ is the length of the shortest cycle contained in $H$. In addition, we consider a family of bipartite graphs with minimum feedback vertex number three. Let $E_{k,t}$ be the graph obtained from the theta graph $\theta_{k,t}$ by joining a new vertex $x$ to one side of the bipartition and another vertex $y$ to the other. Let $E^+_{k,t}$ denote the graph obtained by adding the edge $xy$ to $E_{k,t}$. We prove that for any $k\geq 2$ and sufficiently large $t$, $\text{ex}(n, E^+_{k,t})= \Theta(n^{\frac{3k-1}{2k-1}}).$