Papers

Mathematical programming formulations are routinely modified through algebraic reformulations, perturbations of constraint functions, and the addition or removal of constraints. Some of these modifications preserve the feasible set, whereas others reflect newly imposed modeling requirements and therefore change the optimization problem itself. This paper develops a metric framework for quantifying the robustness of formulation-dependent properties. Formulations are modeled as typed collections of constraint functions. Within each fixed typed-cardinality class, their distance is defined by optimally matching constraints of the same type and measuring the resulting discrepancies in a norm on the underlying function space. As a concrete demonstration of the framework, we study the Linear Independence Constraint Qualification (LICQ). We introduce the pointwise LICQ radius, defined as the distance from a feasible formulation to the set of formulations that remain feasible at a prescribed point but fail LICQ there. Under suitable assumptions, we derive an explicit formula for this radius and characterize the effect of adding a single equality or inequality constraint.

Given a convex polytope $P\subset\Bbb R^d$ with $\mathcal F(P)$ many flags, we prove $$\operatorname{vol}(P) \operatorname{vol}(P-P)^\circ \le \frac{\mathcal F(P)}{(d!)^2}.$$ This implies the following relation between Mahler volume and the number of flag conjectured by Freij, Schmitt, Schymura and Ziegler: for a centrally symmetric polytope $P\subset\Bbb R^d$ holds $$\operatorname{vol}(P)\operatorname{vol}(P^\circ) \le \frac{2^d}{(d!)^2} \mathcal F(P).$$ This shows that the Mahler conjecture implies Kalai's flag conjecture.

We give a deterministic polynomial-time algorithm for the Bilu–Linial signing problem on bipartite graphs. For every finite simple bipartite graph of maximum degree at most an integer $\Delta\ge3$, the algorithm assigns signs to its edges so that the signed adjacency matrix has operator norm strictly less than $2\sqrt{\Delta-1}$. Our algorithm builds on the randomized recursive repair framework of Jadbabaie, Saberi, and Sra~\cite{JSS26}, with deterministic rules for sign selection and vertex deletion.

We give an elementary, straightforward proof of Ovenhouse's interpretation of the denominator and the numerator of a (right) q-deformed rational in terms of weighted lattice paths on a ”snake graph”, and we give an analogous result for the left q-deformed rational introduced by Bapat, Becker, and Licata. The key ingredient is a simple formula for the q-deformation of the matrix of the composition of two homographies of the form z $\rightarrow$ a + 1/z.

We prove that the combinatorial type of a slice of the cube by a hyperplane is completely determined by its graph, within the class of cube slices. We also show that such a result cannot extend to slices of arbitrary simple polytopes by constructing a simple polytope that has two combinatorially distinct slices with isomorphic graphs.

We report on an exhaustive census of all 1,006,700,565 connected graphs on $11$ vertices to find the minimal examples of graphs whose independence complexes are Cohen-Macaulay but not shellable, or shellable but not vertex decomposable. In particular, there are exactly two graphs $G$ where ${\rm Ind}(G)$ is Cohen-Macaulay but not shellable, when the characteristic is not two, and exactly six graphs where ${\rm Ind}(G)$ is shellable but not vertex decomposable.

We develop Delsarte theory for quantum association schemes, allowing both composition and the quantum Schur product to be noncommutative. A MacWilliams identity and positivity of the inner and dual distributions yield conic upper bounds for codes and lower bounds for designs in terms of effective size, which extends subset cardinality. These bounds recover classical linear programs for symmetric association schemes and semidefinite programs for homogeneous coherent configurations. We also quantize Schurian schemes using finite-group representations and characterize quantum block designs in quantum Johnson schemes arising from irreducible stabilizer representations by averaging over subgroups fixing prescribed points individually. In a quantum Johnson scheme on $M_2(\mathbb{C})^{\oplus10}$, we prove that the minimum effective size of a nonzero positive element satisfying the strength-two design condition is $20$, attained by complementary projections outside the center of the underlying algebra. Every such element in the center has effective size $40$, while the conic lower bound is $40/3$, strictly below the actual minimum.

Let $k\ge2$ be a fixed integer. Bollobás and Scott proved that every $3$-uniform hypergraph with $m$ edges admits a partition of its vertex set into $k$ parts such that each part spans at most $m/k^3+O_k(m^{6/7})$ edges. Scott later suggested that the error term should be $O_k(m^{2/3})$. In this paper, we show that every $3$-uniform hypergraph with $m$ edges admits a partition into $k$ parts such that each part spans at most $m/k^3+O_k(m^{2/3})$ edges, thereby establishing the proposed error term.

A collection of $m \geq 2$ lines through the origin in $\mathbb{R}^2$ is balanced if the sum of the orthogonal projections onto these lines equals $m/2$ times the identity. A line arrangement in $\mathbb{RP}^2$, endowed with the round metric of curvature $1$, is point-balanced if the tangent lines at every vertex form a balanced collection. A line arrangement $\mathcal{A}$ in $\mathbb{RP}^2$ has the Hirzebruch property if it consists of $3k$ lines and every line contains exactly $k+1$ vertices. Using a Kempf–Ness convexity argument, we show that an irreducible arrangement $\mathcal{A}$ has the Hirzebruch property if and only if its projective equivalence class contains a point-balanced representative, unique up to orthogonal transformations. We then combine elementary properties of balanced collections of lines in $\mathbb{R}^2$ with spherical geometry to prove that every irreducible point-balanced arrangement is a reflection arrangement. This gives a new proof of Panov's classification of real Hirzebruch arrangements.

Let $\mathcal{G}$ be a finite connected simple graph with at least two vertices. We construct a finite regular hypertope $\Gamma$ whose diagram is the graph $\mathcal{G}$ with every edge labelled by $4$, and for which $Cor(\Gamma)/Aut(\Gamma) \cong Aut(\mathcal{G})$. This shows that any possible group of diagram symmetries can be realized by the correlations of a hypertope. The construction uses a group generated by involutions, with nilpotency class two, exponent four and commutator relations dictated by the defining graph.

We study two graph pebbling parameters, the stacking number and the clearing number, through the Almost Stacked Hypothesis (ASH). This hypothesis asserts that these thresholds can be determined by testing only configurations in which at most one vertex carries more than one pebble. We prove that every almost stacked configuration of size $2^{n+1}-1$ on $C_{2n}$ is stackable and that every almost stacked configuration of size $3\cdot 2^n-2$ on $C_{2n+1}$ is clearable. Together with the known lower bounds, these results show that ASH implies $\operatorname{stack}(C_{2n})=2^{n+1}-1$ and $\operatorname{clear}(C_{2n+1})=3\cdot 2^n-2$. For a finite tree $T$, we introduce an explicit invariant $\operatorname{estim}(T)$. We prove unconditionally that $\operatorname{stack}(T)\geq\operatorname{estim}(T)$ and prove the reverse inequality under ASH. Consequently, ASH yields $\operatorname{stack}(T)=\operatorname{estim}(T)$, and we conjecture that this equality holds unconditionally. Finally, we study perfectly pebblable graphs: finite connected non-bipartite graphs whose clearing number has the minimum possible value $\operatorname{clear}(G)=|V(G)|+1$. Every complete graph with at least three vertices is perfectly pebblable, which might suggest that perfect pebblability requires high edge density. Assuming ASH, however, we show that this is not the case. We give a sufficient criterion involving strong edge-triangulation and Hamiltonian-path and path-cover conditions in vertex-deleted subgraphs and use it to construct two explicit infinite families of perfectly pebblable graphs with edge density tending to zero, one of which has only a linear number of edges.

Let $w(G)$ denote the number of distinct vertex degrees of a finite simple graph $G$. Melnikov asked for a lower bound on the chromatic number $\chi(G)$ in terms of $|V(G)|$ and $w(G)$, and conjectured a strict bound of this type. We resolve Melnikov's valency-variety problem by proving that every graph $G$ with at least two vertices satisfies \begin{equation*} \chi(G)\ge\left\lceil1+\frac{2w(G)(w(G)-1)}{4w(G)(|V(G)|-w(G))+(|V(G)|-w(G)-1)^2}\right\rceil, \end{equation*} and consequently \begin{equation*} \chi(G)\ge\left\lceil\frac{\lfloor w(G)/2\rfloor}{|V(G)|-w(G)}\right\rceil. \end{equation*} Finally, we also construct an explicit infinite family of graphs attaining equality in both bounds. In particular, these examples show that Melnikov's proposed strict inequality is false and that the bounds above are best possible.

Let $f(d)$ be the smallest integer such that every finite connected bridgeless simple graph of diameter $d$ admits a strong orientation of diameter at most $f(d)$. Chvátal and Thomassen (1978) proved $\lceil d^2/2\rceil+d\le f(d)\le2d^2+2d$. In this paper, we prove that $\lceil d^2/2\rceil+d\le f(d)\le\lceil d^2/2\rceil+d+18$ for every $d\ge2$, which shows that $f(d)=\lceil d^2/2\rceil+d+O(1)$ and determines both the quadratic and linear terms up to a bounded additive error. The key method of our proof is to construct a central subgraph $H$ that admits a strong orientation of diameter $O(d)$ and is within distance $\lfloor d/2\rfloor$ of every vertex outside it. We obtain the sharp linear coefficient by jointly estimating outside paths and their connecting paths in $H$ at the actual attachment vertices, rebuilding $H$ when necessary.

An $r$-dynamic coloring of a graph $G$ is a proper vertex coloring in which every vertex $v$ sees at least $\min\{r,d(v)\}$ distinct colors in its neighborhood. The minimum number of colors in such a coloring is the $r$-dynamic chromatic number $\chi_r^d(G)$. We study dynamic colorings of regular graphs. A straighforward observation shows that $\chi_r^d(G)=\chi(G^2)$, for any $r$-regular graph $G$. We prove that $\chi_3^d(G)=\chi(G^2)\le6$ for every claw-free cubic graph $G$, and the bound is sharp. For Hamiltonian claw-free cubic graphs, we improve the bound to $5$ apart from four explicit exceptions. We also determine exactly the $r$-dynamic chromatic number of the $4$-regular circulant graph $C_p(1,3)$ for each $r\in\{2,3,4\}$. In particular, the case $r=4$ determines $\chi(C^2_p(1,3))$.

The switch chain is a Markov chain defined on the set of labelled realizations of a given graphical degree sequence. At each step, a pair of vertex-disjoint edges is chosen at random and the process attempts to replace them with a uniformly chosen perfect matching of the same four vertices, rejecting any proposal that would create a multiple edge. The resulting process is reversible with respect to the uniform distribution on all realizations. We investigate the mixing time of this chain by viewing realizations as the facets of a simplicial complex and studying a variant of the original process called the simplicial switch chain, which we analyze using tools from the theory of high-dimensional expansion. Our technical contributions include a proof that links of faces of sufficiently high codimension are strong spectral expanders and a comparison between the Dirichlet energies of large block updates and two-edge updates. Our main result is an $O(\Delta^{2}m\log m)$ bound on the mixing time of both simplicial and classical switch chains whenever $m\ge C\Delta^{8}$, where $m$ is the number of edges, $\Delta$ is the maximum prescribed degree, and $C>0$ is an absolute constant. For sequences on $n$ vertices with fixed maximum degree, this proves that the chain mixes in $O(n\log n)$ steps, resolving a longstanding conjecture of Cooper, Dyer, and Greenhill and extending its conclusion to irregular degree sequences.

For a graph $G$, let $f_t(G)$ denote the minimum of the maximum degree of an induced subgraph with $\alpha(G)+t$ vertices, where $\alpha(G)$ is the independence number, and write $f(G)=f_1(G)$. Huang's theorem gives $f(Q_k)\ge\lceil\sqrt{k}\rceil$ for the $k$-dimensional hypercube $Q_k$. We extend this lower bound to Cartesian products of $k$ bipartite graphs with perfect matchings and prove that equality holds when the factors are connected and each has a matched bridge. In particular, we prove that $f(T_1\Box\cdots\Box T_k)=\lceil\sqrt{k}\rceil$ whenever each $T_i$ is a tree with a perfect matching. We determine the sensitivity of every Cartesian product of paths, settling the even-path case left open by Zeng and Hou [J. Graph Theory 107 (2024), 169–180]. For these tree products, with $D=\lceil\sqrt{k}\rceil$, we also prove that $f_t(T_1\Box\cdots\Box T_k)=D$ whenever $1\le t\le 2^{D-\lceil\log_2D\rceil-1}$. When $t=2$, this equality holds for all $k\ge 2$, provided that at least one factor is not $K_2$. Matching cuts give an additional exact range for products of even-order paths. Finally, we prove that $f_2(Q_k)=\lceil\sqrt{k}\rceil$ for every $k\ge 2$ with $k\not\in \{4,9\}$, whereas $f_2(Q_4)=3$ and $3\le f_2(Q_9)\le 4$.

How do algebraic relations among unit directions influence the chromatic number of the plane? We introduce monomial Cayley graphs as a framework for investigating this question, beginning with graphs whose allowed unit steps are signed powers of a single complex parameter. Transcendental parameters give bipartite graphs, while for roots of unity we determine both the chromatic and circular chromatic numbers. As a geometric consequence, every graph admitting a unit-distance embedding in the plane in which the angle between any two edge directions is a rational multiple of $\pi$ is three-colorable. For algebraic noninteger parameters, four colors suffice, and three suffice when the primitive integer minimal polynomial has leading coefficient at least three. We conjecture that three colors suffice for every algebraic noninteger direction parameter and that four suffice for every direction parameter.

A $k$-bend path is a non-self-intersecting polyline that lies on a grid and consists of at most $k+1$ axis-parallel line segments. An $\llcorner$-EPG graph is a graph whose vertices can be represented by 1-bend paths on a grid, where each path is either $\llcorner$, $\shortmid$, or $\text{-}$, such that two vertices are adjacent if and only if the corresponding paths share at least one grid edge. Characterizing chordal $\llcorner$-EPG graphs was explicitly posed as an open problem by Cameron, Chaplick, and Hoàng. We resolve this by giving two equivalent characterizations of connected chordal $\llcorner$-EPG graphs. The first assigns to every connected chordal $\llcorner$-EPG graph an ordered clique partition tree of its vertex set, where the underlying tree is the bipartite incidence graph of the horizontal and vertical edge-intersection components. The second is stated purely in terms of maximal cliques: a connected chordal graph is $\llcorner$-EPG if and only if some clique tree admits an admissible partial orientation. A reduction lemma replaces local clique labels by maximal cliques, linking the two characterizations. For a prescribed clique tree, the existence of an admissible partial orientation reduces to 2-SAT and is decided in $O(nM^2)$ time, where $n$ and $M$ are the numbers of vertices and maximal cliques. Finally, we show that strong chordality is not sufficient: we exhibit a strongly chordal graph that is a minimal forbidden induced subgraph for $\llcorner$-EPG, and a family of split graphs with a unique clique tree for which membership in $\llcorner$-EPG reduces to a neighborhood condition.

We study the uniqueness of optimal $1$-embeddings on closed surfaces up to weak equivalence. Answering a question of Suzuki in the negative, we show that for every closed surface other than the sphere, the projective plane and the Klein bottle, there exists an optimal $1$-embedded graph admitting two weakly inequivalent optimal $1$-embeddings. Our construction is given explicitly on the torus and is then extended to other surfaces through a suitable connected-sum construction that preserves weak inequivalence. As a consequence, we also obtain optimal $1$-embedded graphs with exponentially many pairwise weakly inequivalent optimal $1$-embeddings as the genus increases.

Let $G_{n,d}$ be a uniformly random simple $d$-regular graph on $n$ vertices, and let $A_n$ be its adjacency matrix. For every fixed $\lambda\in(0,1/2)$, we prove that for any $\lambda(n-1)\le d\le(1-\lambda)(n-1)$, $\mathbb P(A_n\text{ is singular})\le e^{-cn}$, where $c>0$ depends only on $\lambda$, and $n$ is sufficiently large with $nd$ even.

Kühn and Osthus proved that, for every positive integer $\ell$, every $2^{16}\ell^2$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\geq \ell$ for every $v\in S$. And they asked whether the quadratic bound can be replaced by a linear bound. In this paper, we answer this question in the affirmative by proving that every $641\ell$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\ge \ell$ for every $v\in S$. The proof integrates the Moser-Tardos resampling algorithm.

The $r$th-order Stirling cycle and subset triangles and their associated quasi-Eulerian triangles were introduced by Deb and Sokal in their study of total positivity of combinatorial triangles. They found combinatorial interpretations for the cycle case in terms of Stirling permutations, leaving the subset case open. For $r\ge 2$, we resolve this problem by introducing the notion of Stirling subset permutations along with a consecutive-descent statistic. We also prove the conjectures of Deb and Sokal on the row log-concavity of the higher-order Stirling cycle and subset triangles. Our log-concavity proofs rely on Sagan's criterion, Dey's extension, and strengthened log-concavity inequalities discovered with the assistance of ChatGPT 5.6.

Let $m(p)$ be the least size of a subset of $\F_p$ with at least two elements for which every sum has two distinct representations as unordered pairs, allowing repetition. We prove that, for every prime $p\ge64$, \[ m(p)\ge2^{-80}\left(\frac{\log p}{\log\log p}\right)^2. \] The argument compresses the full integer collision lattice by unit-pivot elimination. A shared random sample and forests of bounded diameter give $O(\sqrt n+n/\log p)$ surviving coordinates of polynomial height for a minimal set of size $n$. A nonzero minor divisible by $p$ then gives the lower bound. We also construct weakly ternary-balanced seeds yielding \[ m(p)\le\frac{(\log p)^2}{2(\log3)^2} +\left(\frac1{4\log3}+o(1)\right) \frac{(\log p)^2}{\log\log p}. \] Consequently $m(p)=(\log p)^{2+o(1)}$ as $p$ tends to infinity through the primes. The constant-factor order of $m(p)$ remains undetermined.

We prove that every cluster monomial of type $A_n$ is log-concave and unimodal with respect to its Laurent expansion in a fan initial cluster. We derive a coefficient formula for laminar interval products arising from compatible diagonals in the polygon model. In even rank, we reduce log-concavity in fan coordinates to log-concavity along integer lines. In odd rank, we use convolution to handle the coefficient sums arising from the fan substitution. This gives partial affirmative answers to the log-concavity conjecture of Chen-Huang-Sun and the unimodality conjecture of Chen. Our result also complements Shen's study on the supports of products of canonical basis elements for cluster X-varieties of type $A_n$.

In this paper, we investigate graphs with the property that all of their powers are well-covered. We obtain complete classifications for several graph classes, including chordal graphs, cochordal graphs, and very well-covered graphs.

We observe a ball bouncing down a Galton board with any number of directions. At each peg it either keeps its direction with some fixed probability or randomly turns to one of the other directions. Its bin measures how often it went each way, a point of a simplex. When the ball rarely turns, the most likely bins are the corners, which only a ball that never turns can reach. We ask when the best bin of each face of the simplex becomes as likely as a corner. To first order every face catches up at the same moment. We break this tie at second order, with an explicit constant for each face. Hence on a long board, as the expected number of turns grows to any fixed multiple of the length's logarithm, the most likely bin jumps once, from the corners straight to the center. A nonuniform start or a weak external field changes the constants, potentially allowing an intermediate face to win.

For every integer $k\ge1$, we prove that each nonempty bounded set in real $\ell_1^{4k+1}$ has an ambient enclosing ball of radius at most $(4k-1)/(4k)$ times its diameter. The bound is sharp in dimension five: $J(\ell_1^5)=3/4$. If a Hadamard matrix of order $4k$ exists, then $J(\ell_1^{4k-1})=J(\ell_1^{4k})=J(\ell_1^{4k+1})=(4k-1)/(4k)$. The upper bound for every $k\ge1$ has been formalized in Lean.

We study the game chromatic number of complete multipartite graphs whose parts are either singletons or are of size at least four. Building on the work of PawełObszarski et al., we give a formula in terms of the ordered part sizes, and the number of singleton parts.

A conference signal is a complex-valued function on a finite abelian group that vanishes at 0 in both the time and frequency domains, and is otherwise flat in both domains. For example, the Legendre symbol is a conference signal on $\mathbb{Z}_p$ since it vanishes at $0$, takes values $\pm 1$ away from zero, and is a scalar multiple of its Fourier transform. More generally, any nontrivial multiplicative character on a finite field is a conference signal on its additive group. Until now, conference signals have mainly been studied by number theorists, who wanted to know if multiplicative characters were the only conference signals on finite fields. The state of the art uses fancy tools from symplectic geometry to prove that non-character conference signals exist on non-small fields. The same tools show conference signals exist on every abelian group of odd order, without saying what they are. In this (human-friendly!) paper, we explain how conference signals produce other objects of interest in algebraic combinatorics and applied harmonic analysis, and then we present some new, explicit constructions of conference signals on $\mathbb{Z}_n$. In particular, we give the first explicit conference signals for composite $n$ and the first explicit non-character examples for prime $n$.

The Double-Partition Coloring Problem (DPCP) is a recently introduced generalization of several coloring problems on graphs and hypergraphs, including the Partition Coloring Problem, the List Coloring Problem, and the Conflict-Free Coloring Problem (CFCP). In this work, we further investigate the structural and computational properties of the DPCP and develop exact and heuristic solution approaches. We derive structural results that allow the detection of infeasible instances and the reduction of their size. We propose one-step and two-step heuristics for obtaining high-quality solutions. These procedures are incorporated into an enhanced branch-and-price algorithm based on a set covering formulation of the DPCP. The algorithm further includes new strategies for solving the NP-hard pricing problem. Computational experiments show that the proposed branch-and-price algorithm and a compact integer programming formulation complement each other: while the compact formulation is generally more effective on sparse instances, branch-and-price performs particularly well on denser and larger instances. In particular, these are the first exact algorithms to be computationally evaluated on various CFCP instances.