Papers

Let $S=k[x_1,\ldots,x_n]$ and let $I=I_\Delta\subsetneq S$ be a nonzero squarefree monomial ideal. Motivated by the classical higher-order Kähler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients $I/I^q$, but more generally the shifted quotients $I^r/I^q$, $1\le r<q$, in the same $I$-adic conormal filtration, together with their symbolic analogues $I^{(r)}/I^{(q)}$. We prove that, for every $1\le r<q$ with $q\ge3$, the module $I^r/I^q$ is Cohen–Macaulay if and only if $I$ is a complete intersection. In sharp contrast, $I^{(r)}/I^{(q)}$ is Cohen–Macaulay if and only if $\Delta$ is a matroid, where loops are allowed. Thus, the Cohen–Macaulayness of a single nonexceptional window forces the Cohen–Macaulayness of every window in the corresponding filtration. The unique exceptional pair is $(r,q)=(1,2)$: at this conormal level, we show that the Cohen–Macaulayness of $I/I^2$ forces $I^2=I^{(2)}$, and hence $I/I^{(2)}$ is Cohen–Macaulay.

We prove that the Enots Wolley sequence contains every positive integer with at least two distinct prime divisors. Suppose, toward a contradiction, that some eligible integer is omitted, and consider its finite set of prime divisors. The local rules then severely restrict how terms involving these primes can occur: after a finite initial segment, terms divisible by some but not all of them can outnumber terms divisible by all of them by at most a fixed constant. A prime-exchange construction gives the opposite conclusion at large scales. From almost every term divisible by all of the chosen primes, it produces enough smaller earlier terms divisible by only some of them; a weighted double count makes this excess quantitative and yields a contradiction. It follows that any omission would force every sufficiently late term to have a prime divisor in one fixed finite set. Prime recurrence and a disjoint-cover argument rule out such a finite obstruction, proving surjectivity. The only analytic number-theoretic inputs are the prime number theorem and Mertens' estimate for reciprocal primes.

Chatterjee and Sloman proved that a bounded measurable similarity function with sufficiently small average Gromov hyperbolicity admits a tree representation with small mean approximation error. Their argument uses a weighted version of Szemerédi's regularity lemma and does not yield explicit quantitative bounds. Here, we establish a tighter relation between average hyperbolicity and mean tree approximation error. For a similarity function $s:S\times S\to[0,b]$, we prove that $$\operatorname{Tree}(s) \leq (63/e)^{1/3} \sqrt[3]{b^2 \operatorname{Hyp}(s)} \leq 2.8512 \sqrt[3]{b^2 \operatorname{Hyp}(s)}.$$ The proof uses a simple pivoting construction inspired by \KwikCluster. We also discuss the optimal dependence on average hyperbolicity, including a square-root lower bound, and connections with ultrametric fitting.

For a real symmetric matrix $H$ and distinct vertices $a,b$, we classify exponents $k$ for which $H^k$ has perfect state transfer (PST) from $a$ to $b$. If their supported eigenvalues are integer multiples of a common positive number, every odd exponent reduces to $H$ and every positive even exponent reduces to $H^2$. We determine the minimum transfer times using a greatest common divisor of supported spectral differences. For rational symmetric matrices, symmetry of the source vertex support about zero implies the odd-power equivalence without a commensurability assumption; this includes all bipartite graphs. If the source vertex supports zero, PST under one positive even power implies PST under every positive even power. For a symmetric three-point quadratic spectrum whose outer projection signs agree and differ from the central sign, a nonzero rational shift leaves exactly one PST exponent. We classify all adjacency powers of hypercubes, cycles, and Johnson graphs, and all adjacency squares of paths. In particular, the adjacency matrix of $P_7$ has PST from vertex $2$ to vertex $6$ only at exponent $2$.

We study the structure of random $d$-regular triangle-free graphs and show that a sharp phase transition occurs at $d=\frac{\sqrt 3}{2}\sqrt{n \log n}$. For smaller $d$, asymptotically almost surely the graph is non-bipartite, whereas for greater $d$, asymptotically almost surely the graph is bipartite.

The representation number of a graph is the smallest integer $k$ such that the graph can be represented by a word in which each vertex appears exactly $k$ times, and two distinct vertices $x$ and $y$ alternate in the word if and only if they are adjacent in the graph. We extend known results on the representation number for various graph classes to chessboard graphs - namely, king, queen, rook, bishop, and knight graphs. We provide a complete classification for queen graphs and partial classifications or observations for the other classes. As a consequence of our study, we obtain a characterization of all chessboard graphs that are circle graphs. Our work also leads to several interesting open problems.

The clique partition number of a graph is the minimum number of complete subgraphs whose edge sets partition its edge set. We study graphs in which, in every induced subgraph and outside every prescribed clique, some vertex has a neighbourhood that becomes a clique after deleting at most $s$ vertices. The case $s=0$ is exactly the class of chordal graphs. For each fixed $s$, we prove that the maximum clique partition number at all sufficiently large orders $n$ is $\lfloor(n+s)(n+s+1)/6\rfloor-\binom{s+1}{2}$, and determine all equality graphs. The same expression is an upper bound up to an additive constant depending only on $s$ at every order. In particular, every chordal graph has clique partition number at most $n^2/6+n/6+O(1)$, answering a question of Erdős, Ordman and Zalcstein. We also prove structural stability for sublinear defect and a sharp finite-order theorem for integer signed clique functionals. The proof combines signed fractional localization with an edge-disjoint triangle construction; only a qualitative fractional-packing approximation is required.

For \(n\ge3\) and a graph \(H\) on at most \(n\) vertices, let \(q(H)\) be the expectation threshold for its containment in \(G(n,p)\). We prove that there are absolute constants \(a,L>0\) such that, for every \(C>0\), every graph of degeneracy at most \(C\log n/\log\log n\) has a deterministic edge decomposition into at most \(\lceil a(C+1)\rceil\) pieces, each with ordinary containment threshold at most \(Lq(H)\). This removes the maximum-degree hypothesis from a theorem of Ascoli, He, Park, and Talagrand in their approach to Talagrand's discrete convexity problem. With constants depending on the fixed parameters, a structural extension allows the addition of \(O(\log n)\) vertices with arbitrary incident edges. The uniform main bound also gives an \(O(1+\log\log n)\)-piece decomposition for every target graph, with an absolute threshold multiplier. The main ingredient converts Li's two-set coupling for spread measures into a partition of a prescribed neighborhood list, fixed before the random host is sampled. It allows arbitrary overlaps and repetitions and controls all Hall matching conditions simultaneously by bounding target demand and host supply through common two-sided approximations under the biased host measure.

For a finite simple graph G, let alpha(G) denote its independence number and let l_avg(G) = (1 / |V(G)|) sum_{v in V(G)} alpha(G[N_G(v)]) be the average independence number of its open neighbourhoods. Written on the Wall II Conjecture 194 asserts that every simple connected graph on n > 1 vertices satisfying alpha(G) <= 1 + l_avg(G) has a Hamiltonian path. We give a four-parameter family of counterexamples. Its principal two-parameter subfamily satisfies the proposed inequality with equality: for every pair of integers s >= 1 and t >= 3 it has (s + 1)t^2 vertices, independence number t + 1, l_avg(G) = t, and minimum degree s, but has no Hamiltonian path. This entire infinite subfamily is machine-checked in Lean 4: one universally quantified theorem certifies its order, connectivity, independence number, average neighbourhood independence, minimum degree, conjecture hypothesis, and failure of traceability. Thus no fixed lower bound on the minimum degree repairs the conjecture. The case (s,t) = (1,3) has 18 vertices, but the formal certificate is parametric rather than a verification of that one graph alone.

For integers $n,r,t$ with $2\le t\le r-1$, let $f(n,r,t+1)$ denote the largest possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and containing no copy of $K_{t+1}$. Lo, Treglown and Zhao identified $f(n,7,4)$ as the only remaining case in their treatment of the $K_4$-free family. I determine this function for every $n\ge1$. First, the corresponding three-colourable extremum $\delta(n,7,3)$ is reduced to a $7\times3$ integer matrix problem and determined exactly. Second, a structural argument shows that every balanced $7$-partite $K_4$-free graph $G$ with $\delta(G)>\frac{132}{31}n$ is three-colourable. Consequently, $f(n,7,4)=\lfloor30n/7\rfloor$ except when $n\equiv4\pmod7$ and $n\ge11$, where it is one less.

The demand matching problem generalizes both the knapsack problem and the $b$-matching problem. In this problem, each edge of a graph has a demand and a weight, and each vertex has a capacity. The goal is to find a maximum weight subset of edges such that, at each vertex, the total demand of the incident selected edges does not exceed the vertex capacity. Parekh [IPCO 2011] proved that, if each edge is individually feasible, the natural LP relaxation for demand matching has integrality gap at most $3$, yielding a $3$-approximation algorithm. This bound is tight for the natural LP relaxation, matching the lower bound of Shepherd and Vetta [Math. Oper. Res. 2007]. We present a randomized $(3/2 + \sqrt{2} + \varepsilon) \approx (2.914 + \varepsilon)$-approximation algorithm for the demand matching problem for every $\varepsilon > 0$, giving the first approximation ratio strictly better than $3$. For bipartite graphs, we obtain a randomized $(2 + \varepsilon)$-approximation algorithm for every $\varepsilon > 0$. Both algorithms run in time polynomial in $1/\varepsilon$ and the input length. Our algorithms use a strengthened LP relaxation based on intersecting the integral knapsack polytopes associated with the vertices, together with a multiple-choice generalization. As a key ingredient, we prove the existence of a $(q, 1/(1+q))$-balanced contention resolution scheme for the integral knapsack polytope for every $q \in [0, 1]$, which may be of independent interest. The balance guarantee $1/(1+q)$ is tight in the worst case over all knapsack instances.

For a constraint satisfaction problem defined by a relation $R$, its non-redundancy $\text{NRD}(R,n)$ is the size of largest instance (as a function of the number $n$ of variables) for which no constraint is implied by the rest. Its chain length $\text{CL}(R,n)$ is the largest such instance where the constraints can be ordered so that no constraint is implied by the preceding ones. Clearly $\text{CL}(R,n) \ge \text{NRD}(R,n)$ but so far no asymptotic separation was known between these quantities. We exhibit an explicit arity $4$ relation for which $\text{CL}(R,n) \ge \omega(\text{NRD}(R,n))$.

Eternal Vertex Cover (EVC) is a turn-based attacker-defender game on an undirected graph $G$. To begin with, the defender places $k$ guards on vertices of $G$. The attacker, on their turn, can choose an edge $e$ not already occupied at both endpoints to "attack". The edge $e$ is defended if a guard moves along the edge $e$. The defender, on their turn, can move any subset of guards. A guard can only move to a neighboring vertex. The minimum number of guards needed to indefinitely defend against any sequence of attacks is called the eternal vertex cover number, generalizing the classic vertex cover number. Determining this number is NP-hard in general, motivating the study of parameterized and approximation algorithms. The problem is known to be FPT when parameterized by the cover number, but structural parameters remain relatively unexplored in the literature. In this work, we explore structural parameterizations for EVC. We show that EVC is FPT parameterized by the cluster vertex deletion number, which generalizes the previously studied parameterization by vertex cover number. We next study the problem parameterized by vertex integrity, which is the smallest number of vertices we need to delete from $G$ so that the resulting graph is a disjoint union of constant-sized components. We first show that Eternal Vertex Cover is XP parameterized by vertex integrity. Then, we develop a polynomial-time approximation algorithm, which computes an additive $6k+1$ ($g(k)$) approximation, where $k$ is equal to the cluster vertex deletion number (vertex integrity). Finally, we show a FPT algorithm for when the deletion set produces "nice" connected components, which are components that are bounded in size and satisfy a technical condition.

We present an exposition of a proof, discovered by GPT-6 Astra, of the Erdős–Sós Conjecture, which states that every graph with average degree greater than $t-2$ contains every tree on $t\geq 2$ vertices.

Answering a question of Noga Alon, we generalize a result of Janzer and Yip to show that every $k$-edge-coloring of $K_{r^k+1}$ contains a monochromatic subgraph with chromatic number at least $r+1$ and diameter $O(k^{3/2}r^{k/2}\log r)$.

The first open case of Hadwiger's conjecture states that every $K_7$-minor-free graph is 6-colorable. We prove that this is the case for $K_7^=$-minor-free graphs, where $K_7^=$ denotes the graph obtained from $K_7$ by deleting two independent edges. The proof is based on an independently interesting density result: Every 5-connected $K_7^=$-minor-free graph with $n\ge 6$ vertices has at most $4n-8$ edges.

We define a diagonal-inversion statistic on labeled Dyck paths carrying both decorated rises and decorated contractible valleys. This gives an explicit candidate for a bivariate refinement of the univariate Theta conjecture of D'Adderio, Iraci, and Vanden Wyngaerd, recently proved by D'Adderio, Pagaria, and the authors of this work. The univariate conjecture can be recovered from our bivariate version by setting $q=1$. The new conjecture recovers the rise and valley versions of the Delta conjecture when either decoration parameter vanishes, and provides a combinatorial interpretation of the symmetric function $\Theta_{e_l} \Theta_{e_k}\nabla e_{n-k-l}$, which is also conjectured to be the Frobenius characteristic of a certain graded module of diagonal coinvariants with two sets of commuting variables and two sets of anticommuting variables. In support of the Theta conjecture and its touching refinement, we prove its Schröder case, that is, the scalar product of the symmetric function side with $e_{n-d} h_d$ matches the combinatorial side restricted to Schröder paths. The proof uses a finer combinatorial argument than the previously known cases, leveraging two Gaussian product identities.

We present an integer-coordinate realization of a genus-3 polyhedral surface with eight planar simple nonagonal faces. It has 24 vertices and 36 edges, and each vertex is incident with three faces. Every pair of faces shares an edge: 20 pairs share one edge and 8 pairs share two. Vertex coordinates, face walks, and plane equations are given. Exact verification confirms that all faces are planar, the surface is closed and orientable, no unintended intersections occur, and the realization has $C_4$ symmetry. The construction is based on an earlier construction and preserves its incidence structure.

Let $G_{n,m}=\overline{\C_n^{[m]}}$, where $\C_n^{[m]}$ denotes the closed $m$th power of the $n$-cycle. We study the graded Betti numbers and homological invariants of the edge ring of $G_{n,m}$ in the range $n\geq 3m+1$. These graphs form a natural family for the study of edge rings whose regularity can be compared explicitly with the induced matching number. In particular, for $n\geq4m+1$, the graph $G_{n,m}$ has induced matching number one, whereas its edge ring has regularity two. Our approach is based on a characterization of the homology of the induced subcomplexes of the independence complex $\Delta(G_{n,m})$. We introduce a family $\mathcal{S}_V(k,m)$ of vertex subsets characterized by their successive gaps around the cycle and show that, for $W\in\mathcal{S}_V(k,m)$, the induced subcomplex $\Delta[W]$ has the homotopy type of $\mathbb{S}^1$, whereas for $W\notin\mathcal{S}_V(k,m)$ all its positive-dimensional reduced homology groups vanish. Combining this characterization with Hochster's formula and an explicit enumeration of $\mathcal{S}_V(k,m)$, we obtain a closed formula for the graded Betti numbers in the second strand. We further determine the extremal Betti number, regularity, and projective dimension of the edge ring of $G_{n,m}$. Finally, we compute the $f$- and $h$-vectors of the independence complex and use the Hilbert series to determine the graded Betti numbers in the linear strand. The case $m=2$ recovers the corresponding results for complements of squares of cycles obtained in~\cite{RatherSquare}.

For any fixed $k$, we prove a lower bound on the $k$th largest modulus of an eigenvalue of the non-backtracking matrix $B$. Specifically, consider any deterministic or random family of graphs that converges locally to the unimodular Galton-Watson tree with root degree distribution $D$, and set $\kappa:=\mathbb E[D(D-1)]/\mathbb E[D]$. Given $\kappa>1$ and an exponential-moment bound on the empirical degree distributions, we show that $|\lambda_k(B)|\geq\sqrt{\kappa}-o_N(1)$, where $N$ is the number of vertices. When restricted to locally tree-like regular graphs, this recovers a well-known consequence of the Ihara-Bass formula. In the specific case where the graph is generated through the Erdős-Rényi model with expected degree $d>1$, this proves a conjecture of Bordenave, Lelarge, and Massoulié. To do this, we show that the normalized log-determinant of the Bethe-Hessian of the graph is bounded by that of the Bethe-Hessian of its local limit. This bound is violated if the eigenvalues of the non-backtracking matrix are too small. We establish this using an effective-conductance interpretation of the tree Green's function recursion.

Sorting By Transpositions (SBT) seeks the minimum number of transpositions required to sort a permutation $\pi$ on $n$ symbols into the identity $\iota$. Let $N=n+1$. A cyclic-target pair $(\omega,\beta)$ consists of an even permutation $\omega$ and an $N$-cycle $\beta$ for which $\rho=\omega\beta$ is an $N$-cycle. An SBT instance is the special case $(\bar{\iota}{\bar{\pi}}^{-1},\bar{\pi})$, where $\bar{\pi}$ and $\bar{\iota}$ encode $\pi$ and $\iota$, and $\bar{\iota}{\bar{\pi}}^{-1}\bar{\pi}=\bar{\iota}$. For a prescribed fixed-point-free cycle type, fixed-content words encode $\omega$, with colors distinguishing cycles and ranks recording their orientations relative to $\beta$. A word is realizable exactly when $\rho=\omega\beta$ is an $N$-cycle. Permutations of equal-part colors and shifts of rank origins form auxiliary symmetries that, together with word rotation and position reflection coupled to rank inversion, define a twisted dihedral action. Its orbits are twisted bracelets, and its realizable orbits correspond bijectively to extended-toric equivalence classes of cyclic-target pairs, where reflection is adjoined to classical toric equivalence. The correspondence yields a Burnside identity, and our method generates one encoding word for each such class. The transposition diameter $TD(n)$ is the largest transposition distance in $S_n$. Combining fixed-point contraction and reductions of the ambient instance space based on cycle structure with exhaustive verification of every remaining twisted bracelet, we prove $TD(16)=9$, closing a twenty-five-year gap. This result also yields $TD(19)=11$ and, for every $n\equiv1\pmod{3}$ with $n\geq16$, $TD(n)\leq\left\lfloor(2n-2)/3\right\rfloor-1$, improving the previous general upper bound by one for these $n$.

Kemeny's constant quantifies the expected time for a random walk to reach a randomly chosen vertex, capturing global properties of a Markov chain. We develop a matrix-analytic framework for bounding Kemeny's constant of a finite connected weighted graph using degree-weighted compressions of the normalized adjacency matrix, pinching inequalities, and eigenvalue interlacing. Our main partition theorem gives lower bounds in terms of the compressed spectrum, with complete equality characterizations, and converts structural graph information into spectrally computable estimates. For instance, when applied to proper color partitions, the proposed method yields a sharp lower bound on Kemeny's constant in terms of the chromatic number \[ K(G)\ge n-2+\frac{1}{\chi(G)}, \] which extends a bipartite bound of Ciardo, Dahl, and Kirkland (2022) to arbitrary chromatic number, and which is incomparable with the normalized Hoffman bound by Chung (1997). For unweighted graphs, we also characterize all equality cases of this bound. We further illustrate the power of the proposed matrix framework by deriving spectral bounds for NP-hard graph problems involving normalized cuts and conductance. Our results include an asymptotically sharp conductance bound, two-sided interlacing bounds from principal submatrices and quotient matrices, and estimates for the change in Kemeny's constant under connectivity-preserving deletion of multiple edges.

Balogh, Bollobás and Narayanan conjectured that among all finite simple $r$-uniform $d$-regular hypergraphs, the number of weak independent sets is maximized by a natural quasi-bipartite construction $H_{r,d}$. We give three types of evidence for this conjecture. For every fixed $r$, we prove the conjectured asymptotic exponential rate whenever the twin quotient has maximum pair codegree $o(d)$. The proof uses the hypergraph container method. For hypergraphs with no cross-edges, the occupancy method gives the sharper error bound $O_r(\log d/d)$. We show that a stronger version of the conjecture in terms of the so-called occupancy fraction is not true, by providing a counterexample for every $r\ge 3$. We also prove exact cases of the conjecture when the hypergraph is $2$-regular. Using an entropy decomposition in the dual edge-cover problem, we settle every odd $r$ and the case $(r,d)=(4,2)$.

There is an absolute constant $c>0$ such that, for every $0<\varepsilon\leq1$, arbitrarily large finite sets $A\subset\mathbb{C}$ satisfy \[ |A+A|\leq |A|^{1+\varepsilon},\qquad |AA|\leq |A|^{2-c\varepsilon},\qquad \nu_1(A)\geq |A|^{1+c\varepsilon}, \] where $\nu_1(A)$ counts unordered unit-distance pairs. We combine the arithmetic directions from the recent unit-distance construction with the multiplicative enlargement used in the recent sum-product construction. The arithmetic ingredients are stated as explicit inputs.

Stanley's conjecture asserts that the $h$-vector of a matroid is a pure $O$-sequence. Corry, Dochtermann, McClain, Perkinson and Yi introduced cycle systems, which give a bijective proof for the matroids that admit one, through coparking functions, and proposed generalized cycle systems. Every matroid admitting a cycle system is regular, and the Wagner and Petersen graphs admit no generalized cycle system consisting of circuits. We propose a relaxation. Where the coparking recursion breaks down, at the strata whose unique union is independent, the coparking functions are replaced by a fibre: a pure multicomplex with the $h$-vector of the dead node, the minor of the matroid attached to the stratum. We prove, for any matroid with a fixed basis, that whenever the required fibres exist the relaxed coparking functions form a pure multicomplex whose degree sequence is the $h$-vector of the matroid, so Stanley's conjecture follows. The proof rests on a version of Dhar's burning algorithm for a matroid with a fixed basis, which gives the purity, and on a deletion-contraction identity that computes the gap between the $h$-vector and the coparking functions as a sum of local $h$-vectors, one per dead node. Coned, biconed and triconed graphs carry a fibre system with their canonical spanning trees, as do the Wagner and Petersen graphs. Beyond graphs, every basis of a matroid of corank two, of a matroid of rank at most four or of a uniform matroid carries a fibre system, so Stanley's conjecture follows for those classes. We also give a graph of radius two on twelve vertices which, with its breadth-first spanning tree, carries no fibre system.

We prove that there is a function $f:\mathbb N\to\mathbb N$ such that $\chi(\Gamma(R))\leq f(\omega(\Gamma(R)))$ for the zero-divisor graph of any ring $R$, if the clique number is finite. On the one hand, this consolidates a disproved conjecture of Beck from 1988, claiming $\chi(\Gamma(R))=\omega(\Gamma(R))$ for unital commutative rings. While previous counterexamples satisfy $\chi(\Gamma(R))\leq \omega(\Gamma(R))+2$, we obtain the lower bound $f\geq k^{\Omega(\log k)}$ among finite commutative rings. Finally, we show that neither zero-divisor graphs of finite commutative semirings nor those of finite commutative nonassociative rings are $\chi$-bounded.

We prove two conjectures regarding partition perimeter inequalities. It was conjectured by Gray, Payne, and Watson that Franklin's partition identity becomes an eventual inequality if one replaces the size of the partition with its perimeter. We prove this conjecture by asymptotic analysis of the corresponding generating functions. Gray, Payne, Swisher, and Watson also conjectured that there is a bias for partitions with fixed perimeter to have more odd parts than even parts. We prove this conjecture by deriving the corresponding generating functions and give a positive recurrence for the coefficients. In the case that there are an equal number of odd parts and even parts we provide a connection to peakless Motzkin paths.

Motivated by Kawanoue's construction of a basis for the derivation module of the cone of the extended Catalan arrangement of type $B_\ell$, we construct an explicit basis for the derivation module of $\mathbf{c}Shi(B_\ell,m)$. Our construction generalizes the previously known case $m=1$ to $m\geq1$. It also provides a partial answer to a question of Abe and Terao concerning explicit constructions of simple-root bases.

Efficient resource use in packing and assembly-line applications requires decisions that jointly account for capacity and precedence constraints. The strongly NP-hard bin packing problem with generalized precedence constraints (BPP-GP) models such decisions by minimizing the number of ordered, capacitated bins required to pack weighted items, even when precedence requirements span multiple bins. Existing exact algorithms primarily focus on classical special cases, whereas general BPP-GP has been addressed only via compact integer models and heuristics, with no efficient open-source exact solver. We present PrecPack, a unified exact solver that extends branch-bound-and-remember (BBR) to arbitrary nonnegative precedence weights and naturally specializes to the classical cases. Generalized states capture restrictions that remain active across future bins, which are addressed through branching, dominance, and conflict-aware lower bounds. Root column generation uses fixed-point arithmetic to compute numerically valid dual bounds for pruning or to prove optimality. To support reuse and verification, we provide common programming and command-line interfaces, independent assignment checking, explicit termination statuses, and reproducible batch execution; the core procedures require no commercial software. In same-machine, single-threaded comparisons on classic assembly-line benchmarks, more instances are proven optimal, and average computing times are substantially reduced relative to leading source-available BBR implementations. Further comparisons with published benchmark results for bin packing with precedence constraints and BPP-GP also show that more instances were proved optimal and that reported average gaps were smaller on most benchmark sets. PrecPack is released under the MIT License at https://github.com/Sunkanghong-Wang/PrecPack.

In 1986, Füredi and Kahn showed that the dimension $\dim(P)$ of any finite poset $P$ satisfies $\dim(P) = O(d \log^2 d)$, where $d$ is the maximum degree of the comparability graph of $P$. Scott and Wood more recently improved this bound to one of the form $d \log^{1+o(1)} d$. We show that $\dim(P) = O(d \log d)$, thus confirming that the corresponding lower bound of Erdős, Kierstead, and Trotter is tight up to the implicit constant.