Papers

In this paper, we establish some curious identities involving Legendre polynomials and the first kind of Apéry-like numbers. As applications, many new supercongruences are deduced.

For a fixed graph $G$, an $r$-uniform hypergraph is said to contain a Berge-$G$ if there exists a bijection $f\colon E(G)\to E(\mathcal{H})$ for some subhypergraph $\mathcal{H}$ such that $e\subseteq f(e)$ for every $e\in E(G)$. Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in $r$-uniform Berge-$K_3$-free hypergraphs with matching number at most~$s$. We determine the exact Turán numbers for the cases $r=3$ and $r=4$. For $r=3$ and $n \geq 3 s$, we prove that every $n$-vertex Berge- $K_3$-free 3-graph with matching number $s$ has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality. For $r=4$ and $n \geq 4 s$, the maximum number of edges is $s\lfloor(n-2 s) / 2\rfloor$, except for the exceptional case $s=1$ and $n \equiv 1(\bmod 4)$, in which the bound is $(n-1) / 2$. As a corollary, our results recover the classical theorem of Győri on Berge-$K_3$-free hypergraphs.

We construct two stable limit representations of the double affine Hecke algebra of type $(C^\vee,C)$ on the space of almost symmetric Laurent polynomials, namely the positive and negative stable limit representations. Starting from the standard polynomial representation of the finite rank DAHA of type $(C_n^\vee,C_n)$, we study the asymptotic behavior of the Cherednik operators under the two natural rescalings by positive and negative powers of the parameter $t$. We prove that these rescaled Cherednik operators admit well-defined limits on the ring of almost symmetric Laurent polynomials. This yields stable positive and negative actions of a common stable limit DAHA. The action of the limit Cherednik operators is also proven to be triangular on a natural basis of almost symmetric Laurent polynomials labeled by tuple-partition symbols with respect to the induced Bruhat order. We further construct for each of the two stable limit representations a set of simultaneous eigenfunctions of the limit Cherednik operators using the partial symmetrization operators acting on the non-symmetric Koornwinder polynomials. We show that each of the two sets of the eigenfunctions form a basis of the space of almost symmetric Laurent polynomials, and denote them by the positive and negative stable limit Koornwinder polynomials.

Hessenberg varieties are subvarieties of the flag variety, defined by containment conditions on flags with respect to a linear operator. The study of these varieties lies in the intersection of algebraic geometry, combinatorics, and representation theory. In this paper, we develop an algebro-geometric procedure for determining the closed subvariety structure of a Hessenberg variety $\mathcal{H}(X,h)$ in the flag variety for any linear operator $X$ and Hessenberg function $h$, by imposing a partial order on the Hessenberg functions and analyzing the relation of the corresponding Hessenberg varieties. In particular, we give a concrete recursive formula for determining all equations cutting out a given Hessenberg variety in each Schubert cell. As an application, we provide an alternative geometric proof of Tymoczko's results on the existence of affine pavings of a given Hessenberg variety and on the dimension count of its cells.

Let $p$ be a prime number. We prove the sharp Furstenberg set bound in the $p$-adic plane $\mathbb{Q}_p^2$: every $(s,t)$-Furstenberg set $E\subset\mathbb{Q}_p^2$ satisfies $$ \dim_H E\ge \min\left\{s+t,\frac{3s+t}{2},s+1\right\}. $$ This matches the sharp lower bound in the Euclidean plane. We also derive two related consequences: a $p$-adic projection theorem for the maps $\pi_\theta(x,y)=x+\theta y$, together with the corresponding exceptional set estimate giving a $p$-adic analogue of Oberlin's projection question; and a discretized fractal sum-product estimate over $\mathbb{Q}_p$, showing that sufficiently non-concentrated subsets of $\mathbb{Z}_p^\times$ cannot have both small sum set and small product set. The proof follows the projection-theoretic and multiscale machinery developed in the Euclidean works of Orponen-Shmerkin (arXiv:2301.10199) and Ren-Wang (arXiv:2308.08819). The main task is to rebuild this machinery in the non-archimedean setting, and along the way we develop several new $p$-adic inputs needed to overcome the ultrametric features of the problem.

Two families \(\mathcal{A}\subseteq\binom{[n]}{k}\) and \(\mathcal{B}\subseteq\binom{[n]}{\ell}\) are said to be nontrivial cross-\(t\)-intersecting if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\), and $|\bigcap_{A\in \mathcal{A}\cup \mathcal{B}}A|<t$. In this paper, we determine the upper bound on \(|\mathcal{A}||\mathcal{B}|\) of two nontrivial cross-\(2\)-intersecting families \(\mathcal{A}\subseteq\binom{[n]}{k}\) and \(\mathcal{B}\subseteq\binom{[n]}{\ell}\) for any positive integers $n,k,\ell$ with \(k\geq \ell \geq 3\) and \(n \geq 3(k-1)\). Moreover, we characterize the extremal families attaining this bound. This settles the last unsolved case of a recent result by He, Li, Wu and Zhang (J. Combin. Theory Ser. A, 217 (2026) 106095).

The character of a connected $\mathfrak{sl}_n$-crystal is a Schur polynomial; the crystal can be further decomposed into quasicrystals, whose characters are the Gessel quasisymmetric functions. Crystal skeletons are obtained by contracting quasicrystals within crystal graphs. They generalize dual equivalence graphs, and can be used to prove the Schur expansion of a symmetric function when the quasisymmetric expansion is known. In this paper, we show that the crystal skeleton can be tiled further into components which we call quasicrystal skeletons, whose characters are Young quasisymmetric Schur functions. We characterize which edges in the crystal skeleton move between quasicrystal skeleton components. Contracting the quasicrystal skeleton components yields Bruhat order. We illustrate how these tools can be applied to symmetric functions by analyzing the Stanley symmetric functions.

We give a reachable-state operator formulation of one-to-one deferred acceptance with strict, possibly incomplete preference lists. A local update operator is defined for each active proposer, while a scheduler selects which local update is applied. On the states reachable from the canonical empty initial state, three invariants are immediate: the set of proposed edges grows strictly, every proposer visits each acceptable receiver at most once, and every receiver holds its most-preferred proposal received so far. These invariants yield termination after at most |E| proposals and stability of every terminal reachable state. The classical rejection lemma then gives proposer optimality and schedule independence. We separate these trajectory statements from two logically different results: the distributive-lattice structure of the full stable-matching set and one-sided strategy-proofness. A diagnostic table records which proof obligation is lost when a model changes the bipartition, ordinal comparisons, proposal irreversibility, or receiver choice rule. The paper makes no new complexity claim; its purpose is a precise operator-level account of the classical proof architecture and of the limits of that architecture.

Conway's 99-graph problem asks whether a strongly regular graph with parameters $\mathrm{srg}(99,14,1,2)$ exists. We develop two complementary lines of attack. Fixing one vertex, the conditions $\lambda=1$ and $\mu=2$ force its neighbourhood to be a perfect matching and determine every edge between that neighbourhood and the remaining vertices. For $(99,14,1,2)$, the unresolved part is therefore a constrained $12$-regular graph on $84$ vertices. We encode this reduction in CP-SAT and validate it by recovering the unique $\mathrm{srg}(9,4,1,2)$. We also prove by exhaustive enumeration that no circulant graph on $\mathbb{Z}/99$ satisfies more than $68.0\%$ of the CAISc constraints, and we give a validated orbit formulation for prescribed automorphisms. We then study the partial-score search problem. Fourteen human-designed search configurations reached at most $69.43\%$. Separately, we supplied the scoring function to an evolutionary program-search system. It produced a degree-preserving $4$-vertex-switch tabu search whose best verified artifact scores $70.73\%$. The generated move differs from those used in our own searches and crosses a plateau that was stable under them. These results do not resolve the existence problem, but they reduce the exact search space and improve the best verified partial construction found in our experiments.

Rashidi, Shajareh Poursalavati, and Tavakkoli [J. Algebra Comb. Discrete Struct. Appl. 7 (2020), no.2, 183-193, Theorem 3.6] claim $Z(P(n,3)) = 8$ for all $n \geq 12$, but this fails at $n = 12$, where $Z(P(12,3)) = 7$. We give an explicit $7$-vertex zero forcing set for $P(12,3)$ and its forcing sequence, and verify by exhaustive search that no $6$-vertex set forces $P(12,3)$. We prove $Z(P(n,3)) \le 8$ for $n \ge 9$ using an explicit forcing set and rotational symmetry. Exhaustive search yields $Z(P(n,3))$ for $7 \le n \le 20$ and identifies the gap in the published proof: its case analysis fails to exclude $7$-vertex sets. We conjecture $Z(P(n,3)) = 8$ for $n \ge 13$, requiring a matching lower bound for all large $n$.

Let \(\mathscr{G}_n\) be the family of simple graphs obtained from \(K_n\) by deleting at most five edges. For a fixed integer \(1\leq k\leq n\), let \(\Phi_k(L(G),x)\) denote the immanantal polynomial of the Laplacian matrix associated with the hook partition \((k,1^{n-k})\). We prove that, for \(n>7\) and \(n\neq 2k-1\), every graph in \(\mathscr{G}_n\) is determined by \(\Phi_k(L(G),x)\) among all simple graphs. We prove that, for \(n>7\) and \(n\neq 2k-1\), every graph in \(\mathscr{G}_n\) is determined by \(\Phi_k(L(G),x)\) among all simple graphs. The proof recovers the order, size, and degree-square sum from the first coefficients, and then separates the remaining candidates by explicit third- and fourth-coefficient comparisons based on the finite classification of complements with at most five edges. The case \(n=2k-1\) is left open because the binomial differences used in these comparisons vanish.

We study the relationship between nowhere-zero three-bit flows, labeled triangles, and cycle double covers of bridgeless multigraphs. Using the flow theorems of Seymour and Tutte, we realize the three flow values at each vertex as the side differences of a triangle whose sides carry two-element subsets of $\F_2^3$. We express agreement of these local labels across graph edges as a binary system and prove its solvability by an inconsistency certificate, a local tester-parity identity, and a global double count. The resulting compatible labels trace cycle components in which every edge occurs exactly twice, proving that every finite bridgeless multigraph has a cycle double cover, proving the cycle double cover conjecture.

For integers $m\ge2$ and $r\ge1$, put $d=m+r-1$ and define the shifted-binomial coordinates $B_{m,r,k}$ by \[ \binom{mn}{d} = \sum_{k=1}^{d+1}B_{m,r,k}\binom{n+k-1}{d}. \] The main purpose of this paper is to identify these coordinates with a specific residue-class decimation of a multinomial coefficient sequence. We prove \[ B_{m,r,k} = [x^{mk-1}](1+x+\cdots+x^{m-1})^{m+r}. \] This identity converts the alternating finite-difference formula for the coordinates into a positive coefficient formula, and yields their exact support. We then prove that the nonzero coefficient vector is palindromic if and only if $r\equiv2\pmod m$. This symmetry criterion is derived here directly within the multinomial-decimation framework. The principal arithmetic results are an unconditional divisibility law and an exact formula for the greatest common divisor of each nonzero row. We prove \[ \frac{m}{\gcd(m,r-1)}\mid B_{m,r,k}, \] and, more precisely, determine \[ \gcd_{1\le k\le K_{m,r}} B_{m,r,k} \] as an explicit product of prime powers determined by the $p$-adic valuations of $m$ and $m+r-1$. Finally, a roots-of-unity filter gives the exact row sum \[ \sum_k B_{m,r,k}=m^{m+r-1}. \] Thus the shifted-binomial coordinates form a positive, arithmetically structured decimation of an $m$-nomial coefficient sequence.

The purpose of this note is to elaborate on the apparent connection between Wronskians and Hessians. More generally, to a given subspace of homogeneous bivariate forms over the complex numbers, we associate two determinantal polynomials called the $W$-polynomial and the $\hat{W}$-polynomial. We give expansion and factorization formulas for these polynomials, and study their behavior under change of coordinates and duality. As an application, we give another proof of Iarrobino's theorem on the strong Lefschetz property for standard graded Artinian Gorenstein algebras in codimension two.

We prove that there exist functions $f:\mathbb N^2\to\mathbb N$ and $g:\mathbb N\to\mathbb N$ such that for all positive integers $k$, $d$, and $\ell\ge3$, every graph $G$ either contains $k$ cycles of length at least $\ell$ that are pairwise at distance greater than $d$, or admits a subset of vertices $X$ with $|X|\le f(k,\ell)$ such that $G-B_G(X,g(d))$ contains no cycle of length at least $\ell$, where $B_G(X,r)$ denotes the ball of radius $r$ around $X$. This generalizes a theorem of Dujmović, Joret, Micek, and Morin (2024), which established the $\ell=3$ case. Moreover, we prove that the theorem holds with $f(k,\ell)\in\mathcal{O}(\ell k\log k)$ and $g(d)\in\mathcal{O}(d)$. The linear bound on $g$ is best possible, while the bound on $f$ is optimal as a function of $k$ for every fixed $\ell$. In particular, for $\ell=3$ our result improves the previous bound of $\mathcal{O}(k^{18}\mathsf{polylog} k)$ by Dujmović et al.

Strongly regular signed graphs are an extension of strongly regular graphs to the realm of signed graphs, that is, graphs where each edge is positive or negative. Unlike with ordinary strongly regular graphs, most kinds of signed counterparts with girth 4 or higher are describable in terms of known structures. We prove that those with girth 4 that are bipartite are classified by designs of two kinds: weighing matrix designs and symmetric block designs. Those of girth 5 are few and readily described. There are none of higher girth. Those with girth 4 that are not bipartite are unsolved.

We prove Kleber's rectangular-complement conjecture for Schur functions over an arbitrary commutative ring $R$, showing that, for a fixed rectangle, the products $s_\lambda s_{\lambda^\vee}$, indexed by unordered complementary pairs, are linearly independent in $\Lambda_R$. The proof rests on a general independence theorem for componentwise splittings, which asserts that for every partition $\theta$, the products $s_\alpha s_\beta$ are linearly independent as $\{\alpha,\beta\}$ ranges over unordered pairs of partitions satisfying $\alpha+\beta=\theta$. The independence of the products $s_\lambda s_{\lambda^\vee}$ also yields linear independence of the Koike–Terada universal-character products over any field, answering a question of Gao–Orelowitz–Yong. We also prove the analogous result for monomial symmetric functions over fields of characteristic zero, as well as integral linear independence over $\mathbb{Z}$.

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, \chi)$ where $T$ is a tree and $\chi : V(T) \rightarrow 2^{V(G)}$ is a function satisfying two axioms: for every edge $uv \in E(G)$ there is an $x \in V(T)$ such that $\{u,v\} \subseteq \chi(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) | u \in \chi(x)\}$ induces a non-empty and connected subtree of $T$. The sets $\chi(x)$ for $x \in V(T)$ are called the bags of the tree decomposition. The tree-independence number of $G$ is the minimum taken over all tree decompositions of $G$ of the maximum size of an independent set of the graph induced by a bag of the decomposition. A graph $H$ is an induced minor of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by vertex deletions and edge contractions. We prove that for every $t\in\mathbb{N}$ there exists an $\epsilon > 0$ such that every graph $G$ either contains the complete bipartite graph $K_{t,t}$ or the wall $W_{t\times t}$ as an induced minor, or has tree-independence at most $O(2^{O((\log n)^{1-\epsilon})})$. This leads to algorithms with running time $2^{n^{o(1)}}$, for a wide range of problems on $\{K_{t,t}, W_{t\times t}\}$-induced minor free graphs. Our result is a substantial generalization of existing bounds for the tree-independence and tree-width on various graph classes, and a partial resolution of the conjecture of Chudnovsky, E S, and Lokshtanov [Arxiv, 2025] that $\{K_{t,t}, W_{t\times t}\}$-induced minor free graphs have poly-logarithmic tree independence number. The generality comes at the cost of a sub-polynomial, rather than poly-logarithmic upper bound. Our result leads to a complete classification of induced-minor closed classes into ones that have sub-polynomial tree-independence, tree-independence equal to $\tilde{O}(\sqrt{n})$, and linear tree-independence.

We study exponentially damped Möbius approximants in $\mathscr H=L^2([1,\infty),dt/t^{-2})$. With \[ \gamma_n(t)=\left\lfloor\frac tn\right\rfloor -\frac{\lfloor t\rfloor}{n},\qquad f(u)(t)=\sum_{n\ge1}\mu(n)e^{-nu}\gamma_n(t),\] we compute the relevant scalar products, characterize the Möbius coefficients as the unique coefficients giving pointwise convergence to the constant function, and prove $\langle1 \mid f(u)\rangle\to1$. Vasyunin's formula expresses $F(e^{-u})=\|f(u)\|_2^2$ as an arithmetic cotangent sum. To analyze $F(x)$ as $x\uparrow 1$, we define the canonical third-order truncation $\mathcal F_{[3]}$ of $F$ by deleting the sole remainder $\rho_3$. We prove exact edge and residue-character cancellations, initial-edge asymptotics, finite-scale formulas, and \[ \mathcal F_{[3]}(x)\ll \frac{\log^2\!\bigl(e/(1-x)\bigr)}{1-x}. \] For the terms containing $\rho_3$, we prove initial-edge asymptotics, and a finite-scale criterion. The unresolved boundedness problem is thereby reduced to explicit global bilinear cancellation.

Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.

A $4$-uniform $2$-cycle in a $4$-uniform hypergraph of length $t$ is a cyclic ordering of $2t$ vertices $v_1v_2\cdots v_{2t}v_1$ such that $v_{2i+1}v_{2i+2}v_{2i+3}v_{2i+4}$ are edges for $0\le i\le t-1$ while the addition is modulo $2t$. For every $\gamma>0$ and large $n$, we characterize the $n$-vertex $4$-uniform hypergraphs such that every triple of vertices is contained in at least $(1/3+\gamma)n$ edges and admits a Hamilton $2$-cycle. Up to the error term $\gamma n$, the assumption on the minimum codegree is best possible and verifies a conjecture of Garbe and Mycroft. As a consequence, this gives a polynomial-time algorithm that decides whether an $n$-vertex $4$-uniform hypergraph with minimum codegree $(1/3+\gamma)n$ contains a Hamilton $2$-cycle. This stands as a steep contrast to the graph case where such a hardness gap has size $o(n)$.

We completely determine the magnitude homology of tope graphs of real hyperplane arrangements. Their ranks can be described as the Hilbert functions of the Stanley–Reisner rings of certain simplicial complexes naturally associated with the arrangements. For Coxeter arrangements, this gives a computation of the magnitude homology of the Cayley graph of the corresponding Coxeter group. We also prove the homological reciprocity for central arrangements conjectured by Koizumi–Liu. The proof combines poset combinatorics, the Edelman–Walker theorem, and Alexander duality.

We give explicit quantum Latin squares of order $6$ with cardinalities $19$, $21$, $23$, $25$, $27$, $32$, and $35$, where vectors differing only by a global phase are identified. Cardinalities $19$ and $21$ arise from symmetric Schur products of columns of dephased Butson matrices. A parameterized direct-sum construction in $\C^6=\C^4\oplus\C^2$ yields cardinalities $23$ and $25$ by a controlled splitting of two pairs of rays. Cardinality $27$ is obtained from mixed Schur products of a $BH(6,6)$ matrix and a row-permuted copy. Two further mixed constructions give cardinalities $32$ and $35$: the first has exactly two three-element phase classes, while the second has exactly one two-element phase class. In every Butson case, Hadamard orthogonality and the ray count are certified by finite arithmetic with exponent matrices. Combined with previously known attainable values and the general impossibility of cardinality $7$, these constructions realize every order-six cardinality except $7$ and the single currently unresolved value $29$.

We prove that all nondegenerate simplices have the canonical Ramsey property, thereby resolving a central open problem in canonical Euclidean Ramsey theory and providing a canonical counterpart to the celebrated simplex Ramsey theorem of Frankl and Rödl~[JAMS, 1990].

For a poset $P$, let $e(P)$ ($e^*(P)$) denote largest positive integer $k$ such that the union of the $k$ middle layers of $2^{[n]}$ does not contain a weak (strong) copy of $P$. Ellis, Ivan, and Leader showed the existence of posets $P$ for which there exists a positive real $\varepsilon_P$ such that $La(n,P)\ge (e(P)+\varepsilon_P)\binom{n}{\lfloor n/2\rfloor}$ and $La^*(n,P)\ge (e^*(P)+\varepsilon_P)\binom{n}{\lfloor n/2}$ hold, where $La(n,P)$ ($La^*(n,P)$) denotes the maximum size of a weak (strong) $P$-free family $\mathcal{F}\subseteq 2^{[n]}$. More precisely, they showed that $P=B_d$ are such posets for all $d\ge 4$, where $B_d$ is the Boolean lattice ordered by inclusion. Tompkins showed that the diamond $B_2$ is also such a poset. We apply his method to settle the case of the last Boolean poset $B_3$. We show that there exists a positive $\varepsilon$ such that $$La^*(n,B_3)\ge La(n,B_3)\ge La(n,D_6)\ge (3+\varepsilon)\binom{n}{\lfloor n/2\rfloor},$$ where $D_s$ is the poset on $s+2$ elements $a<b_1,\dots,b_s<c$. Consider the intervals $I_m=[2^{m-1}-1,2^m-2]$, $I^*_m=[\binom{m-1}{\lfloor \frac{m-1}{2}\rfloor}+1,\binom{m}{\lfloor \frac{m}{2}\rfloor}]$. It is known that for values $s$ in the major initial parts of $I_m$ and $I_m^*$, one has $La(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ and $La^*(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$. The above equalities do not hold for the largest elements of the intervals, thus there exist $s_m\in I_m, s^*_m\in I^*_m$ such that for $s\in I_m$ we have $La(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ if and only if $s<s_m$ and for $s\in I^*_m$ we have $La^*(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ if and only if $s<s^*_m$. Modifying previous constructions, we obtain upper bounds on $s_m$ and $s^*_m$.

We study the interaction between reduction rules and upper-bound functions for the Maximum Clique Problem (MCP). We show how MCP upper-bound functions can strengthen classical core and truss reductions by replacing local size conditions with upper-bound tests. This leads to the \((k,\omega^u)\)-core, the \((k,\omega^u)\)-truss, and the more general \((k,d,\omega^u)\)-truss, where the parameter \(d\) controls the trade-off between stronger reductions and additional computational cost. For each of these notions, we prove clique-preservation properties, correctness of the corresponding peeling algorithm, and running-time bounds. Based on these reductions, we introduce a general framework for improving upper-bound values for MCP. We give two concrete instantiations of the framework: one that uses only the combined truss and core reductions, and one that combines the truss and core reductions with repeated applications of structions. Computational experiments on 73 benchmark graphs show that the proposed reductions can substantially improve several standard upper-bound functions and that combining multiple reduction methods can be beneficial in practice. In particular, the combination of structions, truss and core reductions with a DSatur-based bound often reached SDP-level upper-bound values faster than direct SDP computation; on the tested graphs with edge density below \(0.7\), it did so in every case. Using the truss and core reduction with the Lovász theta upper-bound function, we also improve the previously best certified integer upper-bound values for three difficult DIMACS instances whose exact clique numbers are not known. In particular, we improve upper-bound values for graph C500.9 from 83 to 73, for graph C1000.9 from 122 to 115, and for graph C2000.9 from 177 to 168.

We introduce TITO_Explore, a software package for representing and computing with Translation-Invariant Total Orders (TITOs). We define a canonical window notation for TITOs and design and implement algorithms for several computational tasks involving them. The package normalizes the window notation of a given TITO into its canonical form, computes its inversion set, compares the weak order between two TITOs, and computes the join of two specified TITOs. Our weak order comparison algorithm operates by partitioning the inversion sets into disjoint subsets, thereby breaking down the comparison problem into evaluations of paired subsets. The join algorithm uses an edge-weighted directed graph to represent inversions and converts the problem of finding the join into a weighted path problem in the graph.

Let $P_n$ denote a path on $n$ vertices. A simple finite graph $G$ is called $P_n$-irregular if any two distinct vertices of $G$ belong to a different number of subgraphs of $G$ isomorphic to $P_n$. Alternatively, for a fixed vertex $r$ of $P_n$ (the root), $G$ is called $(P_n)_r$-irregular if any two distinct vertices of $G$ act as the root $r$ in a different number of subgraphs of $G$ isomorphic to $P_n$. This paper proves that for each integer $k \geq 4$, there exists an infinite family of graphs that are simultaneously $P_n$-irregular and $(P_n)_r$-irregular for every integer $n$ satisfying $4 \leq n \leq k$ and every root $r$ of $P_n$. For the path $P_3$, we observe that no nontrivial $(P_3)_r$-irregular graphs exist if $r$ is the central vertex. In contrast, if $r$ is an end-vertex of $P_3$, an infinite collection of graphs is constructed that are both $P_3$-irregular and $(P_3)_r$-irregular. In particular, these results confirm the Strong Conjecture about $F$-irregular graphs for the case where $F$ is a path $P_n$.

We study the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$ and $Z_p$ satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. Within this framework, we investigate the noncommutative binomial formula $(X+Y)^n$ and related identities. In particular, we show how the associated normal ordering coefficients can be expressed in terms of $(p,q)$-deformed $s$-rook numbers. We treat several special cases explicitly, recovering known results from literature as well as deriving new ones.

We introduce the chain replacement of a poset flow: it is obtained by considering the simplicial nerves of the posets of strictly increasing chains in the given poset, ordered by refinement. It maps posets to q-cofibrant flows and inclusions of posets to q-cofibrations. Using the combinatorial properties of the chain replacement, we prove that pushouts along the chain replacement of an order-reflecting inclusion of posets induce trivial h-cofibrations on spaces of execution paths between states of the original flow. By introducing the Hurewicz model structure on flows (or H-model structure), we deduce the same property for any q-cofibrant replacement of an order-reflecting inclusion of posets.