339 papers
This paper establishes that the chromatic number of the random Borsuk graph transitions from being -colourable to requiring more than colours when the average degree is constant for $2 \leq k \leq dk=2$ case via continuum AB percolation.
This paper resolves two major open problems by proving that computing shortest monotone paths on simple polytopes and determining their diameters are NP-hard, while also demonstrating that polynomial-time linear paths can be found on specific small extended formulations of any polytope.
This paper establishes that the chromatic threshold for a homogeneous linear equation over is zero if and only if the equation contains a zero-sum subcollection of at least three coefficients, utilizing equivariant topological methods to resolve a question by Griesmer and extend results on recurrence in infinite abelian groups.
This paper establishes that uniform random triangulations with boundaries in high genus converge locally to half-plane hyperbolic triangulations when rooted at a boundary edge and to the Planar Stochastic Hyperbolic Triangulation when rooted at a random edge, providing a new construction of the former and a proof technique independent of the Goulden-Jackson recurrence.
This paper investigates the division properties of commuting polynomials with rational and integer coefficients, revealing algebraic particularities related to weighted sums on cycle graphs with pendant edges and exploring commuting polynomials over fields of positive characteristic.
This paper determines the percolation and connectivity thresholds for random subgraphs of the permutahedron, a highly-symmetric polytope that generalizes the Erdős-Rényi random graph and the percolated hypercube, while introducing a novel graph exploration technique and initiating the study of the permutahedron's isoperimetric properties.
This paper establishes discrete homotopy and homology theories for finite posets, demonstrating that their discrete and classical homotopy groups are isomorphic and that the two theories are connected via a discrete analogue of the Hurewicz map.
This paper presents new geometrically distinct five-dimensional sphere packings and a nine-dimensional kissing configuration that match existing optimal records without surpassing them, thereby expanding the known constructions of conjecturally optimal arrangements.
This paper establishes closed formulae for the orbifold Euler characteristics of double ramification loci and their higher-rank generalizations in genus one, utilizing a recurrence relation to derive a polynomial result for rank one and a formula involving greatest common divisors of matrix minors for higher ranks.
This paper further investigates the function used to construct a canonical connected fundamental domain for , establishing identities to match its cusps with known classes and detailing the domain's boundary arcs and gluing patterns to aid in understanding the modular curve .
This paper establishes two explicit formulas for counting -points of orbital varieties in ad-nilpotent ideals of type using Hall-Littlewood functions and -weighted tableaux, while providing new proofs and applications for results concerning nilpotent Hessenberg varieties, square-zero matrices, and double coset counts.
This paper constructs two confluent rewriting systems in noncommutative polynomials to establish an algebraic basis for polyzetas, demonstrating that irreducible polyzetas are transcendental and algebraically independent over the rationals, thereby proving that is algebraically independent from odd zeta values.
This paper negatively resolves the question of whether bipartite graphs are well-quasi-ordered by the bipartite minor relation by constructing an infinite set of pairwise incomparable 2-connected bipartite graphs and providing examples that distinguish the bipartite minor relation from the standard graph minor relation.
This paper establishes a spectral stability theorem for -uniform hypergraphs excluding the expansion of a -chromatic graph and applies it to uniquely determine the extremal hypergraph maximizing the -spectral radius among those without vertex-disjoint copies of the expanded complete graph .
This paper establishes that, with specific exceptions, every 3-connected claw-free graph with a domination number of at most 5 is Hamiltonian and every such graph with a domination number of at most 4 is Hamilton-connected, while also proving that 3-connected line graphs of 3-hypergraphs with a domination number of at most 4 are Hamiltonian.
This paper presents a unified method for efficiently computing the generating functions of the Hadamard products between Pell-Padovan tetranacci numbers and eight specific classical second-order recurrence sequences.
This paper investigates the enumeration of permutations avoiding two flat partially ordered patterns by establishing connections to -Fibonacci numbers and restricted permutations to derive generating functions, while also extending existing results on separable permutations with respect to multiple statistics.
This paper employs first-order logic tools to unify and extend existing algebraic and spectral characterizations regarding the isomorphism of controllable graphs and the cospectrality of distance-regularized graphs.
This paper completes the classification of measures on Cameron's elementary treelike Fraïssé classes by establishing a bijection for -colored rooted binary trees that yields infinite families of novel semisimple tensor categories with superexponential growth, while simultaneously proving the nonexistence of such measures on -colored and labeled tree classes for .
This paper demonstrates that any non-adaptive -query relaxed locally decodable code (RLDC) with sufficiently small soundness error can be converted into a standard -query locally decodable code (LDC) with comparable parameters, thereby generalizing previous separation results and yielding improved lower bounds for RLDCs, relaxed locally correctable codes (RLCCs), and probabilistically checkable proofs of proximity (PCPPs).