339 papers
This paper provides combinatorial characterizations of virtually torsion-free and virtually free groups by utilizing canonical graph decomposition theory to establish necessary and sufficient conditions involving the structural properties of -local covers and their associated tree-decompositions.
This paper establishes quantitative bounds on the size of dense subsets of hypercubes that can be embedded into other metric spaces with specific distortion constraints, while also deriving geometric applications regarding nonpositive Alexandrov curvature targets and improving bounds for embeddings of paths and trees.
This paper investigates the limitations of plane substructures in drawings of the -dimensional hypercube graph by constructing drawings with restricted plane paths, matchings, and subgraphs, while also establishing existence guarantees for specific rectilinear configurations and characterizing universal plane subgraphs as forests of caterpillars.
This paper derives explicit formulae for the -dissection of a product of quintuple products when is a prime expressible as a sum of two squares, determines the sign patterns of the resulting Taylor series coefficients, and presents combinatorial applications of these findings.
This paper proves a conjecture by Gyárfás and Király by showing that the vertices of a complete -partite -uniform hypergraph with a spanning -edge-coloring (where ) can be covered by at most monochromatic connected components, while also establishing a related bound for complete bipartite graphs.
This paper characterizes the graphs maximizing the number of -convex sets, establishes the -completeness of counting these sets on split graphs, and provides efficient linear-time algorithms for trees and threshold graphs alongside a novel exact exponential-time approach for general graphs.
This paper determines the minimum total perimeter for an axis-aligned rectangular tiling of a flat torus and proves that this optimal value is always achieved by a tiling consisting of either exactly one or exactly two rectangles.
This paper investigates the set of lattice points formed by pairs of Castelnuovo-Mumford regularity and v-number for edge ideals of connected graphs, establishing general bounds and explicitly characterizing these pairs for whisker and Cameron-Walker graphs while proposing a conjecture for connected chordal graphs.
This paper initiates the study of the hat guessing number under the constraint that the adversary must provide a proper coloring, establishing that complete graphs have a guessing number of $2n-1n \geq 34$, and providing general bounds, exact results for small graphs, and new conjectures for this variation.
This paper reviews the classification of finite distance-transitive graphs, demonstrating that those with large diameters are predominantly geodesic-transitive with structured geodesics, while also providing counterexamples and extending the analysis to polar Grassmann graphs.
This paper establishes quadratic form representations and explicit eigenvalue bounds for the Hessian matrices of resistance distance and Kirchhoff index in positive-weighted graphs using generalized matrix inverses, while also proving the strong convexity of the Kirchhoff index with respect to edge weights.
This paper introduces a general framework for "s-of-k" positional games where players score points by claiming at least elements of a winning set, and comprehensively analyzes the resulting scores under both optimal play and pairing-restricted strategies across various regular grid structures.
This paper initiates the study of minimally tough graphs within the class of weakly chordal graphs by providing complete classifications for several specific subclasses and offering simplified proofs for existing results.
This paper investigates the conditions under which pairwise disjoint curves are unavoidable or avoidable in complete graph drawings, characterizing the resulting plane structures based on specific crossing rules for adjacent and non-adjacent curves.
This paper proves that for sufficiently pseudorandom graphs, the hitting time for the appearance of a Hamilton cycle (and edge-disjoint Hamilton cycles) coincides with the hitting time for reaching minimum degree 2 (or $2k$), thereby resolving long-standing questions by Alon, Krivelevich, and Frieze and establishing sharp thresholds for these properties.
This paper employs the particle motion bijection to establish new -series identities of the Andrews--Gordon type with parity restrictions on part multiplicities, thereby generalizing previous results and providing a simplified proof of a recent identity related to Ariki--Koike algebras.
This paper investigates four relations on the set of point-hyperplane anti-flags, demonstrating that they are generally mutually recoverable except in the specific case of a field with two elements, where one relation fails to recover the others due to a bijection with exterior points of a hyperbolic polar space.
This paper employs cellular sheaves and their cohomology to analyze the infinitesimal rigidity of graph-of-groups realizations, establishing algebraic conditions for Henneberg moves and proving that the Maxwell-count provides a necessary and sufficient condition for minimal rigidity in problems defined over real algebraic groups with specific subgroup constraints.
This paper proves that any family of subsets of an -element ground set containing no -pairwise crossing members has size , thereby confirming the long-standing Karzanov-Lomonosov conjecture regarding the linear growth rate of such cross-free families.
This paper provides a complete combinatorial classification of all modular compactifications of the universal Jacobian over via -functions, characterizing their geometric properties, isomorphism conditions, and structural relationships within an extended stability poset.