Papers

For a graph $G$ and a set $B\subseteq V(G)$, the spread $\mathrm{sp}(B)$ of $B$ is the difference between the largest and the smallest degree in $G$ of a vertex of $B$, and for an integer $k\geq0$ the parameter $\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\mathrm{sp}(B)\leq k$. Caro, Lauri and Zarb derived a lower bound for $\mathrm{sp}(G,k)$ and, among several families of graphs, considered \[ \mathrm{MOP}(n,k)=\min \{\mathrm{sp}(G,k):G\text{ is a maximal outerplanar graph of order }n\} \] and determined $\mathrm{MOP}(n,k)$ up to an additive constant for every $k\not =2,$ leaving the case $k=2$ open, with the bounds $4n/9\leq \mathrm{MOP}(n,2)\leq (5n+19)/11$. We first prove a lower bound on $\mathrm{sp}(G,k)$ for an arbitrary graph $G$ in terms of its order $n$, its number of edges $m$ and its minimum degree $\delta $. This lower bound contains the bounds of Caro, Lauri and Zarb and, for $k=0$, the bound $\mathrm{rep}(G)\geq \left\lceil n/(2d-2\delta +1)\right\rceil $ of Caro and West, where $d=2m/n$. We determine when this lower bound is attained, exhibit explicit graphs attaining it, and show that it is exact for all graphs once $n\geq n_{0}(\delta ,k,d)$. We then apply the bound to maximal outerplanar graphs: adjusting the count to this class we prove \[ \mathrm{MOP}(n,2)\geq \left\lceil \frac{4n+10}{9}\right\rceil \qquad \text{for every }n\geq 14, \] with equality for $n\equiv 2\ (\mathrm{mod}\ 18)$, and $\mathrm{MOP}(n,2)=4n/9+O(1)$ for every $n$.

We prove that the integral Chow groups $CH^i$ and mixed Hodge degree $H^{2i, (i, i)}$ cohomology groups of really full rank (RFR) sink-recurrent cluster varieties vanish for $i > 0$. In particular this applies to braid varieties and open Richardson varieties in any Lie type. Our main tool is the construction of a stratification of any RFR sink-recurrent cluster variety $\mathcal{A}(\Sigma)$ into (affine spaces times) RFR sink-recurrent cluster varieties of seeds with fewer mutable vertices than $\Sigma$. We employ the theory of Voevodsky motives, and towards this end we prove that the cycle class maps are isomorphisms onto the lowest-weight part of rational Borel-Moore homology for any mixed Tate variety over a number field. We then show that RFR sink-recurrent cluster varieties have mixed Tate and, in fact, split motives. Finally, we use our results to deduce vanishing theorems about the Khovanov-Rozansky homology groups of closures of positive braids and generation properties of the cohomology of closed Richardson, projected Richardson, and brick varieties.

Circulant digraphs form a prominent class of Cayley digraphs defined on finite cyclic groups. Building on the existing classification of $2$-arc-transitive circulant graphs, this paper presents a complete classification of $2$-distance-transitive circulant digraphs. Our main theorem establishes that every connected $2$-distance-transitive circulant digraph is isomorphic to one of the following: the undirected cycle $C_n$, the complete bipartite graph $\K_{\frac{n}{2},\frac{n}{2}}$, the complete multipartite graph $\K_{m[b]}$ with $m\geq 3,b\geq 2$, the graph $\K_{\frac{n}{2},\frac{n}{2}}-\frac{n}{2}\K_2$ for odd $\frac{n}{2}$, prime-order Paley graphs, the directed cycle $\overrightarrow{C}_n$, the oriented graph \(G(p^m,r)\) satisfying Condition~\ref{p-power-normal-2dt-cond}, the oriented graph \( C_r(b,1)\) with $r\geq 3,b\geq 2$ and $rb=n$, the lexicographic product oriented graph \( G(p^m,r)[\overline{\K}_d]\) where \(G(p^m,r)\) obeys Condition~\ref{p-power-normal-2dt-cond}.

We prove that a path maximizes the expected range of a uniformly chosen graph homomorphism into the integers, with one vertex pinned at zero, among all connected bipartite graphs of the same order. This establishes the expectation form of the Benjamini–Häggström–Mossel conjecture. The proof restricts and rescales a homomorphism on each bipartition class, then contracts the edges on which the resulting height function is constant. A quantitative estimate for the rank of these zero edges compensates for a parity term in the expected range of a simple random walk, allowing an induction on the number of vertices. We then prove that the BHM inequality implies the Loebl–Ne\v set\v ril–Reed inequality for uniformly chosen integer 1-Lipschitz functions on arbitrary connected graphs, and hence obtain the LNR conjecture as a corollary of BHM. The proof was obtained through interaction with OpenAI GPT-6 Astra and verified by the author. The main results have also been formalized and checked in Lean~4.

We give an efficient algorithm with improved algorithmic guarantees for the (offline) Beck–Fiala problem when the sets have bounded size. Let $A$ be an arbitrary matrix $A\in\{0,1\}^{m\times n}$ with at most $d$ ones per column and at most $s$ ones per row. Let $\log^*$ denote the iterated logarithm and $\ell_j$ denote the $j$-fold composition of log. Assume $s\le\exp(O(\sqrt d))$. We provide an efficient algorithm that, for arbitrary sparsity $d$, gives $O(\sqrt d(1+\log^*n))$ discrepancy. Moreover, if $d\ge\ell_j(n)$ for a fixed integer $j\ge1$, the algorithm gives $O_j(\sqrt d)$ discrepancy. The proof is a bootstrapping scheme using the Bansal-Jiang algorithm.

In this paper, we study polyhedral aspects on commutative and noncommutative Edmonds' problems for computing the rank of linear symbolic matrix $A = \sum_{k=1}^m A_k x_k$. We regard them as linear optimization over integral polytopes ${\cal P}(A)$ and ${\cal Q}(A)$, respectively, which are obtained by the convex hulls of exponent vectors of subdeterminants of~$A$ and its blow-ups $A^{\{d\}} = \sum_{k=1}^m A_k \otimes X_k$ $(d=1,2,\ldots)$. By extending previously known results on nc-rank, we establish a hierarchy of integral polytopes ${\cal P}(A) \subseteq {\cal P}^{\leq 2}(A) \subseteq {\cal P}^{\leq 3}(A) \subseteq \cdots = {\cal Q}(A)$ and show that the integrality gap of ${\cal Q}(A)$ relative to ${\cal P}(A)$ is at least $1/2$. Further, we show that if each $A_k$ is rank-2 skew-symmetric, then the above hierarchy terminates at the second level and the integrality gap is improved to $2/3$.

For a connected graph $G$, let $\tau(G)$ denote the maximum number of pairwise edge-disjoint spanning trees, and let $\rho(G)$ be its adjacency spectral radius. For integers $n\ge1$, $k\ge2$, and $k\le\delta\le2k-1$, let $\mathcal{G}_{n,\delta}$ be the class of connected $n$-vertex graphs with minimum degree $\delta$, and let $\mathcal{L}_{\mathcal{H}}^{2}(n,k,\delta)\subseteq\mathcal{G}_{n,\delta}$ be the exceptional family introduced by Chang, Li, and Zhang. For sufficiently large $n$, the sharp adjacency-spectral threshold for $\tau(G)\ge k$ is determined by maximizing $\rho(G)$ over this family. Set $h=\delta-k$. For each fixed admissible pair $(k,\delta)$ and all sufficiently large $n$, we determine the maximum and minimum adjacency spectral radii in $\mathcal{L}_{\mathcal{H}}^{2}(n,k,\delta)$. In the core–placement representation, let $M$ be the missing-edge graph of the bounded core. For $h\ge2$, the unique maximizer, up to isomorphism, satisfies $M\cong K_{1,h}\cup(h+1)K_1$, with the exceptional edges nested on the large-clique side. Hence the matching configuration proposed in Conjecture~2 of Chang–Li–Zhang is not extremal. For every $h\ge1$, the unique minimizer satisfies $M\cong hK_2\cup2K_1$, with the $2h$ exceptional edges having distinct endpoints on the large-clique side. We also settle the cases $h=0,1$ and all equality cases. The proof combines an exact core–placement parametrization with a uniform Schur-complement resolvent expansion. The first candidate-dependent coefficient is affine in $\sum_{x\in V(M)}d_M(x)^2$, while the first placement-sensitive coefficient is a squared-load functional. Equitable quotient matrices yield the two extremal radii, and the maximizing graph gives the sharp global adjacency-spectral threshold.

Recently, Ferroni constructed a family of counterexamples to the well-known conjecture in Ehrhart theory stating that the $h^*$-polynomial of a lattice polytope with the integer decomposition property is unimodal. This raises the question of whether the $h^*$-polynomial of a lattice simplex with the integer decomposition property remains unimodal. In this note, we prove that every lattice simplex with the integer decomposition property and prime normalized volume has a unimodal $h^*$-polynomial. Furthermore, we establish several sufficient conditions for the unimodality of the $h^*$-polynomial of such simplices.

For ordered graphs $H_1,\ldots,H_t$, let $\rt(H_1,\ldots,H_t)$ denote the least integer $N$ such that every $t$-coloring of the edges of the naturally ordered complete graph on $[N]$ contains an ordered copy of $H_i$ in color $i$ for some $i\in[t]$. We prove that a uniformly random ordered matching $M$ on $n$ vertices with interval chromatic number two asymptotically almost surely satisfies \[ \rt(K_3,M) =\Omega\left(\frac{n^{4/3}}{(\log n)^{1/3}}\right). \] This strengthens the lower bound $\Omega((n/\log n)^{5/4})$ of Balko and Poljak for such random matchings and improves the general existential lower bound of Conlon, Fox, Lee and Sudakov by a factor of $\log n$. The proof combines pseudorandom triangle-free graphs, a coarse encoding of order-preserving embeddings, and a permutation avoidance estimate derived from Brègman's inequality.

The Kemeny rule aggregates multiple strict rankings into a single strict ranking that minimizes the sum of its distances from the input rankings. The resulting optimization problem, called the Kemeny problem (KP), is a special case of the linear ordering problem (LOP). The Kemeny rule satisfies several desirable properties in social choice theory, including the extended Condorcet criterion (XCC). Ando et al. strengthened this result by introducing the strong Condorcet criterion (SCC) and showing that it holds for every optimal solution to an arbitrary LOP instance. Yoo and Escobedo extended the Kemeny rule to rankings with ties and showed that the resulting rule satisfies the non-strict extended Condorcet criterion (NXCC). This criterion gives a condition under which one candidate must be ranked strictly above another in every optimal solution. In this paper, we introduce the non-strict strong Condorcet criterion (NSCC), a counterpart of the SCC for rankings with ties, and show that it holds for every optimal solution to an arbitrary instance of the linear ordering problem with ties (LOPT). We also establish a complementary structural property that gives conditions under which two candidates must be tied in every optimal solution to an arbitrary LOPT instance.

A peeling sequence of a finite planar point set is an ordering of point removals, in which each point is a vertex of the convex hull of the points not yet removed. Write $g(S)$ for the number of such sequences, and $g(n)$ for the minimum of $g(S)$ over $n$-point sets in general position. We present a method which can be used to prove better upper bounds on the previously analysed recursive 3-branch constructions $S_n$. In fact, we prove $g(n)\leq g(S_n)=O(6.57^n)$, using directional restrictions and a weighted prefix-tree argument.

Croft, Falconer and Guy ({Unsolved Problems in Geometry}, Problem~C5) exhibit a tiling of the $27\times27$ square by eight pairwise incomparable integer rectangles and remark that it is not known whether $27$ is the smallest side length of a square that can be tiled by pairwise incomparable integer rectangles, no restriction being placed on the number of tiles. We show that it is: for every integer $n\le26$ and every $k\ge2$, the $n\times n$ square admits no tiling by $k$ pairwise incomparable integer rectangles. The proof combines two structural reductions with an exhaustive search over the $167\,538$ surviving candidate tile sets, carried out by two independently written programs. The complete software, build instructions and output logs are included as ancillary files.

Local-to-global techniques for establishing spectral gaps have played a central role in the modern theory of Markov chain mixing times and the theory of high-dimensional expanders. One of the most striking results in this burgeoning literature is that a spectral gap for the global down-up walk on the facets of a pure simplicial complex can be reduced to sufficiently strong spectral expansion of just the codimension-2 links of the complex, a phenomenon colloquially referred to as "trickle-down". These types of theorems have had many important applications, including rapid mixing of the exchange walk on the bases of any matroid. In this primarily expository article, we give streamlined proofs of two such theorems in the literature, one by Oppenheim (2018) and one by Leake and Oveis Gharan (2025), via an integrated Bochner method. Moreover, in the latter setting, we quantitatively strengthen the dependence of the global spectral gap on the dimension of the complex and the spectral influence, resolving an open question of Leake and Oveis Gharan. Disclaimer: The proofs were developed through a couple of rounds of interaction with GPT-5.6 Sol Ultra. We later discovered that Guo and Zhang (2026) had independently proven the same strengthening of the trickle-down theorem of Leake and Oveis Gharan using an extremely similar argument, also found by GPT-5.6 Sol Ultra. The focus of their paper is the complexity of approximating the partition function of spin systems on planar graphs, not on the trickle-down phenomenon itself. In contrast, our motivation is primarily expository, and we hope to bring Bochner-type methods and their connections with the trickle-down phenomenon to the attention of a wider community of researchers.

A unit ball is surrounded with a ring of $n$ equal infinite circular cylinders. Each cylinder touches the ball and its two neighbors. Can one more cylinder of the same radius be added if the configuration may be rearranged? Can the $n$ cylinders instead be enlarged, and by how much? We prove that insertion is impossible for $3\le n\le7$ and possible for $n\ge9$. The case $n=8$ remains open. The case $n=6$ is Kuperberg's problem from 1990, which was resolved recently. We prove that the ring of four cylinders cannot be expanded. This settles a conjecture of Ogievetsky and Shlosman from 2018. For $n\ge5$ the largest possible expansion radius remains unknown.

We prove that every equal-weight spherical $4$-design on $\mathbb{S}^2$ has at least twelve points. Since the regular icosahedron is a spherical $5$-design, this determines the exact minimum$$N_4(\mathbb{S}^2)=12;$$equivalently, no such design has $9$, $10$ or $11$ points.The proof is part of a finite-defect theory. If a spherical $2m$-design has corank $c=N-\dim P_m$, its Naimark complement consists of unit vectors $u_x\in\mathbb{S}^{c-1}$ forming a spherical $2$-design and satisfying the exact coupling$$u_x\cdot u_y=-\frac{K_m^{(d)}(x\cdot y)}{c}\qquad(x\ne y).$$This gives a pairwise kernel bound, an antipodal lower bound, Cayley-Bacharach information, and uniform lower bounds for multiplicative-relation spaces. In corank one the design splits into two equal spherical $m$-designs. We derive a residue formula for its signed Schoenberg coefficients; the coefficient of degree $m+3$ is negative exactly when $3\le d\le m+1$, excluding corank one throughout that range. At strengths four and six the only examples in any dimension are the regular hexagon and octagon, respectively.In corank two the complement is a circle Gale frame. Multiplication by its phase forces at least $d-1$ linear-quadratic aliases and yields exact norm and socle identities in every dimension. For eleven nodes on $\mathbb{S}^2$, two aliases produce a real harmonic cubic and a Hermitian quartic matrix. A matrix-valued Cayley-Bacharach argument eliminates the generic branch; the exceptional branch reduces to a Pauli normal form and contradicts the second moments. In dimensions $d\ge4$ the corank-two problem remains open; we identify a forced quadratic socle as the obstruction to extending the present argument.

A majority C-colouring partitions a graph into classes in which every vertex has at least half of its neighbours. Write $M(G)$ for the maximum number of classes. We determine $M(K_q^{\square(2k+1)})=\left\lfloor\frac{q^{k+1}}{\lfloor q/2\rfloor+1}\right\rfloor\qquad(q\ge3,\ k\ge0).$ The lower bound follows from explicit rectangular partitions and a uniform three-dimensional bridge. A punctured-rectangle construction supplies both the bridge and a partition in the intermediate dimension. The upper bound is a classical consequence of Hamming edge isoperimetry; an elementary proof is included. We also prove $M(C_m\square P_n)=\frac n2\lfloor m/2\rfloor$ for $m\ge4$ and even $n\ge2$, contradicting the cylinder assertion of Conjecture 4 in arXiv:2608.27669v1 for odd $m\ge7$ and even $n\ge6$. Further layer bounds determine additional cylinder and torus families.

Let $f_{0,1,4}(n)$ denote the number of partitions of $n$ into parts congruent to 0, 1 or 4 modulo 5 with each part used at most twice, and let $f_{0,2,3}(n)$ be defined analogously for parts congruent to 0, 2 or 3 modulo 5. We prove that for every prime $p \equiv 3 \pmod 4$ there are explicit non-negative integers $a(p)$ and $b(p)$, determined by $20a(p) \equiv -9$ and $20b(p) \equiv -1 \pmod{p^2}$, such that $f_{0,1,4}(p^2m + a(p))$ and $f_{0,2,3}(p^2m + b(p))$ are congruent modulo 3 to $f_{0,1,4}(m)$ and $f_{0,2,3}(m)$ when $p \equiv \pm 1 \pmod 5$, and to $f_{0,2,3}(m)$ and $f_{0,1,4}(m)$ when $p \equiv \pm 2 \pmod 5$. By quadratic reciprocity the two functions are preserved exactly when 5 is a quadratic residue modulo $p$, and interchanged otherwise. The smallest cases are $f_{0,1,4}(9m) \equiv f_{0,2,3}(m)$ and $f_{0,2,3}(9m+4) \equiv f_{0,1,4}(m)$. The proof reduces each generating function modulo 3 to the square of a Rogers-Ramanujan-type theta function by means of $(1-x)^{\ell} \equiv 1-x^{\ell} \pmod{\ell}$ and the Jacobi triple product, and then dissects the resulting binary quadratic form using the fact that $-1$ is a quadratic non-residue modulo $p$. As corollaries we obtain, for each such $p$, a congruence with exceptions on a progression of modulus $p$, and an infinite family of self-similarity congruences on progressions of modulus $p^{dn}$ with $d = 2$ or $d = 4$ and constants $9(p^{dn}-1)/20$ and $(p^{dn}-1)/20$.

For an infinite product P(q) = prod_{m>=1} (1-q^m)^{eps(m)} and real delta, write P(q)^delta = sum_{n>=0} c_delta(n) q^n. Schlosser and Zhou conjectured precise sign patterns for these coefficients, for several products and ranges of delta. We resolve four of their conjectures completely, in each case extending partial results already in the literature: those for the Goellnitz-Gordon product Q_8 = (q,q^7;q^8)_inf/(q^3,q^5;q^8)_inf (Conjecture 21), for Q_12 = (q,q^11;q^12)_inf/(q^5,q^7;q^12)_inf (Conjecture 24), and for the Borwein products G_7 and G_11 (Conjectures 20 and 23). Conjecture 23 is true. Conjecture 21 fails exactly on [beta, 8/3), where beta is approximately 2.66448; the threshold 8/3 is sharp, and the first counterexamples occur near n = 7.6 x 10^5. Conjecture 24 fails exactly on (delta_1, 0), where delta_1 = -0.64411... is a root of the 22nd coefficient polynomial. Conjecture 20 fails exactly on (delta_c, 5), where delta_c = 4.8735075853867342634... is a root of the 897th coefficient polynomial. All other stated ranges are proved. Parts of Conjecture 21 were settled independently by He and Li; what is new here is the range near delta = -1, where their threshold diverges. Each failure has the same mechanism: a leading circle-method amplitude vanishes, and a secondary term with the wrong sign overtakes it. The proofs are self-contained apart from classical facts. The cusp analysis is exact, via finite orbits of Weil representations; the Hardy-Ramanujan-Rademacher expansion is made fully explicit; and certified computations in exact and ball arithmetic handle the finite ranges and the degenerate regimes.

Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. The tensor product $G \times H$ (also known as the direct product or Kronecker product) is one of the four standard graph products. Problem 7.2.5 in Kitaev and Lozin's book Words and Graphs (Springer, 2015) raised three open questions regarding the word-representability of tensor products. Despite extensive research on word-representable graphs, the word-representability of tensor products seems to have received no attention. This paper not only answers 2.5 of the questions posed by Kitaev and Lozin in Problem 7.2.5, but also initiates a systematic study, focusing on four fundamental families: wheel graphs $W_n$, complete graphs $K_n$, the Mycielskian of the cycle graph $\mu_n$, and the extended Mycielskian of the cycle graph $\mu'_n$. Among our main results, we prove that $W_{2n} \times G$, $\mu_{2n} \times G$, and $\mu'_{2n} \times G$ are always word-representable for any graph $G$; that $K_n \times K_m$ is word-representable if and only if $\min\{n,m\} \leq 3$; and that tensor products $G \times H$ containing $W_{2n+1} \times W_{2m+1}$ or $\mu_{2n+1} \times \mu_{2m+1}$ or $\mu'_{2n+1} \times \mu'_{2m+1}$ as induced subgraphs are non-word-representable. Our proofs exploit the hereditary nature of non-word-representability and the presence of non-comparability neighbourhoods.

The Total Coloring Conjecture (TCC) is a challenging unsolved problem posed by Behzad and Vizing independently, which states that every simple graph $G$ admits a ($\Delta(G)$ +2)-total-coloring, where $\Delta(G)$ denotes the maximum degree of $G$. This conjecture has been confirmed for graphs with $\Delta(G)\leq 5$. However, for planar graphs, the only open case is $\Delta(G)=6$. It was known that planar graphs with maximum degree 6 and without 4-cycles are 7-totally-colorable. In this paper, we improve this result by showing that any planar graph $G$ of maximum degree 6, which does not contain some special 4-cycles, is 7-totally-colorable.

This paper continues the study of the $K$-theory of monoid schemes, using it to give a useful definition of the higher $K$-theory of a matroid via its Bergman fan.

We show that binary codes of length $n$ correcting two deletions exist with redundancy $3\log_2n+O(\log_2\log_2n)$. The previous best upper bound had leading coefficient $4$, unchanged since 1965, while the best known lower bound has coefficient $2$. More generally, codes correcting $t\ge2$ deletions exist with redundancy $(2t-1)\log_2n+O_t(\log_2\log_2n)$, improving the coefficient $2t$. We extract a code from one label class of a random linear hash of substring counts, with about $n$ times fewer labels than a direct construction. Confusable words that still share a label are separated by a two-colouring after discarding the words in components with odd cycles, and these are few because an odd cycle forces the edits along it to overlap.

In recent work, Hirschhorn and the second author defined $a_k(n)$ to be the number of partitions of $n$ wherein the even parts come in only one color, while the odd parts may be “colored” with one of $k$ colors for fixed $k\geq 1$. This function generalizes the classical partition function and has been of significant interest because it satisfies a number of congruences. Although prior work studying $a_k(n)$ has resulted in congruences in arithmetic progressions in a somewhat ad hoc manner, this work gives a uniform framework for studying congruences modulo 3. This allows us to prove an infinite family of infinite families of non-nested congruences modulo 3.

Consider a sequence of i.i.d.~trials $X=\{X_1, X_2, \ldots, X_n\}$ where $p(X_i=j)={1}/{d}; j=1,2,\ldots, d$, or more generally $p(X_i=j)=p_j; \sum_{1\le j\le d}p_j=1$. We consider the variable $D$ that counts the number of distinct substrings of all lengths,$1\le k\le n$ in $X$ and prove results concerning $E(D)$.

The Pachter-Sturmfels phylogenetic rank of a graph G is the minimal number of metric trees needed to embed G isometrically. Here, all edges of G are of length one and the product of metric trees is endowed with the supremum norm. We develop both a greedy and an exact algorithm for computing phylogenetic ranks. Using our algorithms, we construct a database of phylogenetic ranks which includes all graphs on 6 and 7 vertices. In particular, we exhibit examples disproving that the phylogenetic rank is hereditary, bounded by $\lceil \frac{n}{2} \rceil$, and a generalised 4-point conjecture by Pachter and Sturmfels. In addition, we show that the phylogenetic rank is subadditive under 1-sums and certain 2-vertex-sums, and that it is trivially upper bounded by n-1, where n denotes the number of vertices. We also provide a complete classification of graphs with phylogenetic rank 1 and construct several infinite families with phylogenetic rank $\lceil \frac{n}{2} \rceil$.

Posets of integer partitions, ordered by refinement, were criticized for unruly structural behavior by both Birkhoff and Ziegler. However, recent work of Propp, McConville, and Sagan demonstrated that such posets of hyperbinary partitions are distributive, with posets of join irreducible elements isomorphic to the well-studied class of fence posets. We provide a generalization of this result for hyper $(b,t)$-ary partitions, where all parts are powers of some positive integer $b$, and multiplicities are restricted to $t\ge b$. Then, we show that posets of hyper $(b,t)$-ary partitions are indeed also distributive lattices.

We prove that there is an absolute constant $c>0$ such that for every finite $(\alpha,M)$-Ahlfors–David regular set $A\subseteq[N]$ with $0\leq\alpha<1$ and $M\geq2$, we have the sumset estimate \[|A+A|\geq |A|^{1+c(1-\alpha)/\log M}.\] We also prove a corresponding statement for Ahlfors–David regular sets in $[0,1]$. The proof combines a multiscale entropy decomposition with inverse results from additive combinatorics, showing that Ahlfors–David regularity forces a definite entropy gain at each scale. The dependence $1/\log M$ in the exponent is of optimal order.

A graph $G$ is called $F$-irregular if all its vertices have distinct $F$-degrees, defined as the number of subgraphs of $G$ isomorphic to a given graph $F$ and containing the respective vertex. We prove the Strong Conjecture about $F$-irregular graphs (Dovzhenok, Filuta, and Chuhai, 2024), which states that for every connected graph $F$ of order at least three, there exist infinitely many $F$-irregular graphs. Fundamentally generalizing the classical existence conjecture by Chartrand et al. (1987), this work presents an authorized English translation of our original February 2024 manuscript, which was publicly presented in full at two scientific conferences the same year.

Counting the number of coverings of $s$ dimers on two-dimensional quadratic lattices is considered as intractable and belongs to \#P-complete class. We reveal the structure of the exact solution to the problem and provide an explicit formula for it, which includes $s-1$ nesting sums. This results in an exponential time complexity of $O(2^s)$. The solution is explicitly determined by a sequence that exhibits double-exponential growth.

We study the monoid $\mathcal W$ of based closed walks on the $1$-skeleton of the three-dimensional cube $Q_3$. The fundamental group of this graph is the free group $F_5$. We prove that every additive one-step reading, given by a sum of edge weights in an abelian group, factors through the directed transition counts, and we exhibit two closed walks with equal transition counts and different reduced loop words. Hence commutative aggregation does not determine the reduced loop word. The reduced loop word gives a proper noncommutative recognition quotient. We also determine the shortest closed walk with trivial abelianization but nontrivial degree-two commutator information. Its minimum length in the cube edge metric is $14$. We also obtain a separation between finite and unbounded memory. A two-state reading separates an order pair, while for every $k\geq1$, there is an explicit factorial pair which no $k$-state reading separates. An unbounded stack recovers the reduced loop word on every walk. For integer-valued functions on the vertex set, potential readings vanish on closed walks, while occupation binding is not rectangular. More generally, every occupation-based constraint is rectangular on a class of histories if and only if the occupation vector is constant on that class. Finally, a declared quarter-turn quaternion clock gives a second proper order-sensitive congruence, incomparable with the reduced-word quotient. The obstruction to commutative recovery, and the minimum $14$, remain valid on every hypercube $Q_n$, $n\geq3$.