Papers

Fang and Lin [J. Algebraic Combin. 63 (2026), Art.~58] asked whether, whenever $F$ is edge-color-critical with $\chi(F)=r+1$, every non-$r$-partite, $F$-free graph of maximum adjacency spectral radius must also maximize the number of edges. We give a negative answer. Let $F=K_1\vee\mu(K_3)$, where $\mu(K_3)$ is the Mycielskian of a triangle. This graph is edge-color-critical with $\chi(F)=5$. We prove that $\SPEX_{5}(n,F)\cap\EX_{5}(n,F)=\varnothing$ for all sufficiently large $n$. Thus no graph can simultaneously maximize both the edge count and the spectral radius.

We prove the equivalence of Hindman's finite sums theorem with a natural extension of Ramsey's theorem.

We first study the part size statistic in cylindric partitions, listing some specializations of a crude but general formula. Then, we establish bijections between ordinary partitions and cylindric partitions of small profiles with additional restrictions. Lastly, we extend earlier formulas for generating functions of cylindric partitions into distinct parts and functional equations of unrestricted cylindric partition generating functions, involving the part size statistic.

Let $G$ be a graph of order $n$ with adjacency matrix $A_G$, let $\mathbf e$ denote the all-one vector, and let$W_G=[\mathbf e,A_G\mathbf e,\ldots,A_G^{n-1}\mathbf e]$ be its walk matrix. We consider the case $\operatorname{rank}W_G=n-2$, the first corank for which distinct graphs can have the same walk matrix. We give a complete structural description of such pairs. More precisely, if $G$ and $H$ are distinct graphs on the same labelled vertex set and $\operatorname{rank}W_G=n-2$, then $W_G=W_H$ if and only if $H$ is obtained from $G$ by a reciprocal Wang–Qiu–Hu (WQH) switching. In this case, $A_G-A_H=uv^T+vu^T$, where $u,v\in\{0,\pm1\}^n$ have disjoint supports and form a basis of $\ker W_G^T$. We also determine the minimum order at which a non-isomorphic corank-two walk mate can occur: no such pair exists for $n\le9$, so the previously known $10$-vertex example is sharp. Starting from a labelled realization of that pair, we use singleton union and join operations, together with the graph coronal, to construct connected non-isomorphic pairs with equal corank-two walk matrices for every $n\ge10$.

We study Ehrhart coefficients, the integer decomposition property, and unimodular triangulations for two families of Hermite normal form simplices considered by Bruckamp, Caicedo, and Juhnke. The first family consists of simplices of the form $S_{\boldsymbol{a}}=\mathrm{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol{a})$, where $\boldsymbol{a}=(a_1,\ldots,a_{d-1},N)$, and the second consists of the simplices $T_{d,N}=\mathrm{conv}\bigl(0,e_1,\ldots,e_{d-2},(d-2,\ldots,d-2,d-1,0),(1,\ldots,1,N)\bigr)$. For the family $T_{d,N}$, we prove unimodality of the Ehrhart coefficients in every dimension and determine the log-concave and real-rooted cases. For the subfamily $S_{\boldsymbol{a}}$ with $\boldsymbol{a}=(N-q,\ldots,N-q,N)$, we characterize both the integer decomposition property and the existence of a regular unimodular triangulation via a congruence condition on a negative continued fraction. We also give an arithmetic test for the integer decomposition property and finite triangulation tests for $S_{\boldsymbol{a}}$ with arbitrary $\boldsymbol{a}$. These results address three open questions of Bruckamp, Caicedo, and Juhnke.

Covers in hypergraphs are frequently studied to capture various forms of dependence between hyperedges. For example, even covers–which check if each vertex appears in an even number of hyperedges–have found much success recently in the study of locally decodable codes. Inspired by a recently-emerging line of work on the non-redundancy of constraint satisfaction problems (CSPs), we introduce and study two novel families of covers of hypergraphs which are stricter than even covers: Abelian covers and Catalan covers. Abelian covers are similar to even covers, except that arithmetic is now done over the integers rather than modulo 2, allowing us to capture dependences over arbitrary Abelian groups. Catalan covers capture the behavior of non-Abelian groups by only allowing local cancellations in a sequence of hyperedges. We prove three main results about Abelian and Catalan covers. First, using tools from lattice theory, we show that any $r$-uniform hypergraph with $n$ vertices and $n \log(r)$ hyperedges has an Abelian cover. Second, using tools from algebraic topology, we show that in any $3$-uniform hypergraph, Abelian covers and Catalan covers are equivalent; thereby showing that Catalan covers emerge after $O(n)$ hyperedges in $3$-uniform hypergraphs. Finally, using the theory of nilpotent groups, we show that there exists a $4$-uniform hypergraph which has an Abelian cover but not a Catalan cover. Collectively, these results exactly characterize the reach that Abelian covers have in deducing dependences in hypergraphs. As our primary application, we show that any arity-$3$ CSP with an infinite-domain Mal'tsev extension has linear non-redundancy. This implies near optimal streaming, sparsification, and kernelization algorithms for this family of CSPs. Previously, such a result was only known for the much simpler case of arity-$2$ CSPs.

We prove two arithmetic nonexistence conditions for tight spherical $5$-designs in dimension $(2m+1)^2-2$. Let $m$ be a positive even integer satisfying $\nu_2(m(m+1))\leq5$, where $\nu_p(a)$ denotes the exponent of the prime $p$ in the positive integer $a$. We show that such a design does not exist if either (i) $m\equiv1\pmod3$ and $\nu_p(m(m+1))\leq1$ for every prime $p\geq7$ with $p\equiv3\pmod4$, or (ii) $m\equiv0\pmod3$, $\nu_3(m(m+1))=1$, and $\nu_p(m(m+1))\leq1$ for every prime $p\geq5$ with $p\not\equiv1\pmod{12}$. Together, these conditions recover the corresponding nonexistence results of Bannai, Munemasa, and Venkov and of Nebe and Venkov, and exclude parameters not covered by either earlier result, including $m=16,40,88,96,100$. The proofs use lattice theory to constrain the discriminant groups of the associated lattices, and derive contradictions through Gauss sums and determinant congruences.

Let $G$ be a finite simple graph and let $A_c(G)$ be the Artinian algebra associated with its cover ideal. We prove that $A_c(G)$ has the WLP when $\tau(G)>|V(G)|/2$, where $\tau(G)$ denotes the size of a minimum vertex cover of $G$. As a consequence, $A_c(G)$ has the WLP with high probability when the Erdős-Rényi random graph model is considered. Moreover, we study the borderline case $\tau(G)=|V(G)|/2$ and as a result, classify the WLP for paths, cycles, Ferrers graphs, and well-covered trees.

Let $A=(a_{ij})$ be an $n\times n$ real matrix and let $\lambda$ be a partition of $n$. Let $\phi^\lambda$ be the class function dual to the Young permutation character, and let $$ \phi^\lambda[A] = \sum_{\sigma\in\mathfrak{S}_n}\phi^\lambda(\sigma) \prod_{i=1}^{n}a_{i\sigma(i)} $$ be the corresponding monomial immanant. Stembridge [Canad. J. Math. 44 (1992), pp. 1079-1099] posed the following open problem: If all minors of $A$ of order at most $r$ are nonnegative, and the partition $\lambda$ has length at most $r$, is it true that $\phi^\lambda[A]\ge 0$? This paper solves the problem for all hook partitions.

The classification of block-transitive 5-designs remains an open problem. The additional parameter condition that $k$ divides $v$ is called the Camina-Gagen condition. In this paper, we investigate block-transitive simple $5$-$(v,k,2)$ designs satisfying the Camina-Gagen condition. Using the classification of finite $2$-homogeneous permutation groups, we consider the affine and almost simple cases separately. We prove that no such design admits a block-transitive automorphism group of affine type. For the almost simple case, up to isomorphism, there are exactly two possibilities: a $5$-$(12,6,2)$ design admitting PGL(2,11) as a block-transitive automorphism group and a $5$-$(24,8,2)$ design admitting PGL(2,23) as a block-transitive automorphism group.

Jackson and Thomassen conjectured that every $2k$-strong digraph contains a spanning $k$-strong oriented subdigraph. We prove sharp degree conditions for the existence of such a subdigraph. For every fixed positive integer $k$ and all sufficiently large $n$, every $n$-vertex digraph $D$ with $\delta^0(D)\ge\lfloor(n+k-1)/2\rfloor$ admits a $k$-strong orientation. This threshold is best possible even for the weaker conclusion that $D$ itself is $k$-strong. We also prove a sharp Woodall-type analogue for every fixed positive integer $k$ and all sufficiently large $n$: if $d_D^+(x)+d_D^-(y)\ge n+2k-2$ for every missing arc $xy$, then $D$ admits a $k$-strong orientation, and this bound is again best possible. As a further consequence, we determine the sharp minimum total degree threshold. Finally, the semi-degree result also remains valid when $k\le\alpha n$ for every fixed $0<\alpha<0.094882\ldots$.

We study the distribution of $A_\alpha$-eigenvalues of a tree. We extend the works \cite{Jacobs2021} and \cite{SIN2020} by proving that the number of $A_\alpha$-eigenvalues, for $0\leq \alpha \leq \frac{1}{2}$, less than or equal to the average degree $d_\alpha= \alpha (2- 2/n)$ is, at least, $\lceil \frac{n}{2} \rceil$ for a tree with $n$ vertices. Several counterexamples for $\frac{1}{2} < \alpha \leq 1$ are exhibited. We also derive the same result for other well-known families such as the Deformed Laplacian and the matrix $B_\beta$.

We give a self-contained elementary proof of the known result that every set $A\subset\R$ of cardinality seven satisfies $|A+A|\le |A-A|$. The argument adapts Hegarty's method, using representation counts and the largest positive differences. An exact counting identity, two applications of the Cauchy–Schwarz inequality, and an analysis of a symmetric six-element set with one point added complete the proof, with no computer enumeration. This answers in the affirmative a question of Chu for the seven-element case.

We prove every term of Svinin's even-order recurrences is a Laurent polynomial in the initial values.

Binomial coefficients for multidimensional arrays count occurrences of particular configurations. For the sake of presentation, the emphasis is put on column-binomial coefficients for two-dimensional finite arrays. In this article, we first show that these coefficients can be computed through some Magnus transform. To get structural and combinatorial information on arrays, we then define \((k,\ell)\)-binomial equivalence for finite arrays and, from it, the \((k,\ell)\)-binomial complexity function of an infinite array. Roughly, two finite arrays are \((k,\ell)\)-binomially equivalent when they share the same number of subarrays of size at most $k\times \ell$. We obtain general results on the \((k,\ell)\)-binomial complexity of infinite arrays coding direct products of infinite words. Our main theorem gives an exact formula for the \((k,\ell)\)-binomial complexity of the two-dimensional Thue–Morse array. To that end, we closely examine the action of the bit-wise complement on the \(k\)-binomial equivalence classes of the factors of the Thue–Morse word.

We show that all normal covers of a fixed finite graph $G$ sitting between the universal and maximal abelian covers share the same atomic part of their spectral measures. We give an explicit double formula for the mass of atoms which generalises and unifies known results on universal covering trees and maximal abelian covers. It also proves an extension of a conjecture in physics literature on hyperbolic lattices. Moreover, we also establish the logarithmic Hölder regularity of the continuous part of the spectral measures. Our proof combines techniques from von Neumann algebras, matching theory and the monotone labelling method. In an appendix, we prove a converse generalised Gallai-Edmonds structural theorem which might be of independent interest.

The order $m$ antisymmetrised matrix product ensemble is represented by the product $X_1^T\cdots X_m^TJX_m\cdots X_1$, where $X_1,\ldots,X_m$ are independent real Ginibre matrices and $J$ is the elementary antisymmetric matrix, while the order $m$ Hermitised matrix product ensemble is represented by $X_1^\dagger\cdots X_m^\dagger HX_1\cdots X_m$, where $X_1,\ldots,X_m$ are now independent complex Ginibre matrices and $H$ is a Hermitian matrix drawn from the Gaussian unitary ensemble. These ensembles have recently been shown to be related to certain Muttalib–Borodin ensembles and integrals of Harish-Chandra–Itzykson–Zuber type, thereby motivating further investigation into their eigenvalue statistics. In this work, we construct ribbon graphs and constellations that are enumerated by the mixed cumulants of these ensembles and give loop equation characterisations for the generating functions of said cumulants when $m=1$.

We introduce a new family of recursively constructed sparse matrices, termed Fractus matrices, and investigate their use in constructing low-density parity-check (LDPC) codes. Generated through a self-similar recursive process, these matrices yield regular sparse parity-check matrices while preserving key structural properties across successive iterations. This recursive structure enables an efficient encoding algorithm with computational complexity that is nearly linear in the block length. Decoding is performed using standard iterative message-passing algorithms, thereby retaining the low-complexity decoding characteristic of LDPC codes. The proposed construction produces Tanner graphs with girth six and guarantees a minimum Hamming distance of at least $\ell+1$. We establish several algebraic properties of Fractus matrices, including sparsity, regularity, recursive decomposition, and symmetry under the flip-transpose operation. In addition, we show that the family of Fractus matrices admits a natural lattice structure and that the associated LDPC codes inherit corresponding lattice-theoretic properties. These results establish a connection between order theory and coding theory. Overall, the proposed framework integrates recursive matrix constructions, efficient encoding, graph-theoretic analysis, and lattice theory into a unified algebraic approach to the design and analysis of scalable LDPC codes.

We prove a dimensional strengthening of the colorful Helly theorem. Let $\mathcal{C}_1,\dots,\mathcal{C}_{d+1}$ be finite nonempty families of convex sets in $\mathbb{R}^d$. If $C_1\cap\cdots\cap C_{d+1}\neq\varnothing$ for every choice of $C_i\in\mathcal{C}_i$, then $\sum_{i=1}^{d+1}\dim(\bigcap\mathcal{C}_i)\geq0$, where $\dim\varnothing=-1$. We also obtain a dimensional strengthening of a theorem of Kim and Lew, in which the intersections are taken over unions of color classes. For simplicial complexes, we introduce the link-Leray dimension, defined as the Leray number of the link for a face and as $-1$ for a nonface. Using this invariant, we prove a matroidal extension of the dimensional colorful Helly theorem for $d$-Leray complexes. We further establish a corresponding strengthening of the topological Kim–Lew theorem in two ranges of parameters. As a special case, we recover the topological colorful Helly theorem of Kalai and Meshulam. Finally, we construct counterexamples to the unrestricted topological extension, even under a stronger local condition.

The classical Hadwiger-Nelson problem asks for the chromatic number of the unit distance graph of the Euclidean plane. Over the years, this problem has been considered for a variety of other normed spaces, with higher-dimensional Euclidean space $\mathbb{R}^d$ being perhaps the most natural and well-studied. Alon, Bucić, and Sauermann proved that, for a typical norm on $\mathbb{R}^d$, the chromatic number of the unit distance graph is at most $2^d$. We improve this exponential bound to a linear one by showing that the chromatic number of a typical norm on $\mathbb{R}^d$ is at most $2d$ and that this is tight. This shows a stark difference in the behavior compared to the Euclidean case, where there is an exponential lower bound. One of the key ingredients is a certain high-dimensional, matrix generalization of the Lonely runner conjecture, which also allows us to completely settle the so-called view-obstruction conjecture of Schoenberg from 1978 and a more recent covering-radius conjecture of Henze and Malikiosis.

For a graph $H$, $\langle H \rangle$ denotes the class of all subdivisions of $H$ and $tw(H)$ denotes the treewidth of $H$. In this paper, we prove the following. For $k\geq 1, R\geq 1$, let $G,H$ be two graphs such that strong isometric path complexities (Chakraborty et al. [Disc. Math., 2026]) of both $G$ and $\langle H \rangle$ are at most $k$, and $G$ admits an honest, “nicely rooted” $R$-bounded $H$-partition. Then, there is a graph $F$ with $tw(F)\leq tw(H)$ such that $G$ admits a $(1,33\cdot R\cdot k^2)$-quasi-isometry to $F$. Using results of Albrechtsen, Distel, and Georgakopoulos (2025), we also obtain that $K_{2,t}$-asymptotic minor-free graphs admit quasi-isometries with additive distortion to $K_{2,t}$-minor-free graphs. This answers an open question raised by the above authors. As part of our proof, we combine the graph-partition based method and the layering partition based method (Chepoi et al. [Discrete Comput. Geom. 2012]) to obtain additive quasi-isometry when both the source and all subdivisions of the target graph have bounded strong isometric path complexity.

The algebra of polysymmetric functions (PSym) is defined as the tensor product of copies of the algebra of symmetric functions (Sym) where the $i$th copy has variables scaled by $i$. One way to construct a basis of PSym is to start with a basis $\{f_\lambda\}$ of Sym and consider all pure tensors arising from this basis. Asvin G and Andrew O'Desky described four families of non-pure tensor bases, namely $H, E, E^+$ and $P$, which we call plethystic bases. In this paper, we study the expansions of the plethystic bases into the pure-tensor bases $\{h^\otimes_\tau\}$ and $\{e^\otimes_\tau\}$, and interpret the results via combinatorial objects called polywrapping block tabloids. The expansions of $H, E$ and $E^+$ in $h^\otimes$ and $e^\otimes$ are found via the $h$-expansion of the plethysm $h_n[p_r]$. We compute this $h$-expansion using a statistic on words and prove it through abacus methods.

We give a short proof of the following theorem of K. Ball: If a centrally symmetric convex body $K\subset \mathbb R^d$ is covered by finitely many planks, then the sum of their relative widths with respect to $K$ is at least $1$.

We study Verlinde sums $V_{n}(N,m)$, which for our purposes are finite sums of trigonometric functions $\{r_{x}\}$, associated to $\textrm{ SU}(N)$. In this article we prove the following results: (i) the polynomial $D_N(w):=\prod_x(w-r_x)$ is equal to, up to an explicit and computable factor, the iterated resultant of $Q(u)=u^m-1$ with an explicit elementary polynomial $W_N$ which is independent of $n$ and $m$; (ii) for fixed $N$ and $m$, $V_n(N,m)$ satisfies a linear recursion of order at most $\binom{m-1}{N-1}$, together with arguments showing that one actually has a recursion of length equal to the number of distinct roots of $D_{N}$; (iii) for fixed $N$ and $m$, a Binet-type closed form expressing $V_n(N,m)$ as an explicit finite sum of $n$-th powers of algebraic numbers which are rescaled reciprocals of the distinct roots of $D_N(w)$ is derived. Several explicit examples are provided for $N=2,3,4,5,6$ and $8$.

In DNA-based storage systems, data are retrieved by sequencing molecules sampled from a DNA pool. We study how the way coding redundancy is shared between files affects their expected retrieval times. Our focus lies on the case of two files that are encoded by a systematic linear code over an arbitrary finite field. We consider the conjecture that the sum obtained by dividing each file dimension by its expected retrieval time is at most one whenever the dimension of at least one file is more than one. For this, we develop a geometric view of the retrieval process. As molecules are sampled, we follow the growing span of the corresponding columns of the generator matrix and track how much of this span comes from each file. Among the samples that enlarge the overall span, this lets us compare those that make progress toward recovering both files with those that make progress toward neither. We call a column mixed if the corresponding encoded symbol combines information from both files. In this work, we sharpen a projection bound and use it to control the effect of mixed columns. We prove the conjecture whenever the total information dimension is at least twice the number of mixed columns plus two. For equal-sized files, this allows up to one fewer mixed column than the dimension of either file and extends the previous result for codes with no mixed columns.

Let $d \geq 5$, $0 < \gamma < d - 2$, and $\Omega_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-\gamma}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq \Omega_N$ of fixed positive relative density realizes, at each scale $p_N^{-2/(d - 2)} \lesssim \lambda \lesssim N^2$, a squared distance of the form $q^2 \lambda$, where $q$ belongs to a fixed finite set depending only on the dimension and the density. The lower scale $p_N^{-2/(d - 2)}$ is sharp. As a consequence,the squared-distance set $D^2(B)$ of $B$ has maximal order $N^2$ and contains affine copies of every fixed finite subset of $\mathbb Z$. The main new input is a finite multidilate supersaturation theorem for dense subsets of $Q_N$, which, together with boundedness estimates for the associated spherical distance graphs down to the sharp scale, allows us to apply Schacht's transference theorem.

Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph $\Gamma_E(R)$ and the associate class graph $\Gamma_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. This count is a graph invariant, and we estimate it for $\Gamma(R)$, $\Gamma_A(R)$ and $\Gamma_E(R)$. We prove that the count for $\Gamma_A(R)$ is bounded below by the count for $\Gamma_E(R)$, and that the lower bound is attained precisely when $R$ is Gorenstein. As a consequence we obtain that $R$ is Gorenstein if and only if $\Gamma_A(R)\cong\Gamma_E(R)$ as graphs, the isomorphism being an arbitrary one and not merely the natural compression map. Using the same counting technique we then answer, for Artinian rings, a question of Anderson and LaGrange by showing that $\Gamma(R)\cong\Gamma_E(R)$ if and only if $R\cong \mathbb Z_2^{\,n}$ for some $n\ge2$, or $R\cong\mathbb Z_4$, or $R\cong\mathbb Z_2[x]/(x^2)$.

We study one-dimensional binary cellular automata with a five-slot radius-two local rule of exact algebraic-normal-form degree two, acting on periodic rings of length n. We prove that every such rule is non-injective whenever 4 | n and n >= 8. The proof begins with the four-cell collapse, where the two extreme formal slots coincide. A structural classification of the resulting four-variable maps separates the 65,472 exactly quadratic rules into 63,456 rules with an immediate ring-four collision and 2,016 exceptional lifts. The latter split into layers of sizes 480 and 1,536. Their remaining finite obligations are represented by 136 parameter-region constructions and 768 per-lift records, respectively. Each certificate supplies differentiating closed walks of lengths 8 and 12 with a common pair-graph base vertex. Concatenation then gives lengths 8a + 12b, which are exactly the multiples of four from eight onward. The load-bearing finite certificate core therefore contains 904 = 136 + 768 independently replayable objects checked by standalone, non-searching programs. The complete checker CLIs additionally reconstruct expected populations and execute coverage, complement, and regression/guard checks; 904 is not a count of total checker operations. Periodic extension also yields full-shift non-injectivity; that consequence is used here only as a corollary.

Fishburn's latent-subset conjecture, proposed in 1987 and revisited in 1988, asserts that for every dual intersecting family $\mathcal F\subseteq 2^{[n]}$, there exists $i\in[n]$ such that $|\mathcal F^L(i)|\geq|\mathcal F(i)|$. Here $\mathcal F^L$ is the family of the subsets of members of $\mathcal F$ that do not belong to $\mathcal F$. In this paper, we prove the conjecture using the recent weighted star inequality of Chang, Liu, and Liu.

We show that if an $(n,d,\lambda)$-graph satisfies $\lambda\leq (1-\delta)d$ and $d\gg \delta^{-6}(\log n)^{3}$ for some $\delta>0$, then it is Hamiltonian. A qualitatively similar result was recently proven by Bradač and Janzer. Our proof here is shorter and gives better quantitative bounds. \par In our proof, as in earlier work of Ferber, Han, Mao, and Vershynin, we use a random matrix inequality to show that a mild spectral gap is typically preserved after randomly sampling an appropriate proportion of the vertices. This allows us to deduce that typical balanced bipartite subgraphs of pseudorandom graphs contain perfect matchings. To convert a collection of perfect matchings into a Hamilton cycle, we use a variant of the sorting network method.