Specialty
A mathematical preprint proves that a disjoint-clique construction gives the exact maximum number of large cliques in broad families of graphs that avoid a fixed tree. The result moves the reported general threshold to logarithmic order but leaves small clique sizes, especially triangles, open.
Scientific publication5 min read
A computational preprint reports 84 improved upper- and lower-bound cases for q-ary covering codes across 83 parameter cells, with exact checks on codes and certificates but several unresolved gaps.
Scientific publication4 min read
A theoretical study builds regular graph families that satisfy expansion conditions yet have no cycle covering more than a small fraction of all vertices. The constructions do not settle the same question for robust expanders.
Scientific publication4 min read
A mathematics preprint maps a repetition-based exponent for infinite words, finding every value from 1 to infinity over three symbols and a probability-one value of 1 under Bernoulli measure.
Scientific publication3 min read
A new counting construction improves the rigorous lower bound for the growth rate of permutations avoiding the pattern 1324, while leaving the exact growth rate unknown.
Scientific publication3 min read
Every qualifying pair of tournaments has an intersection discrepancy on the n to the three-halves scale, with related density-dependent bounds for oriented graphs and fixed families.
Scientific publication4 min read
A mathematical preprint sets a vertex limit for minimal forbidden graphs, identifies the complete forbidden set at a threshold of 3, and narrows the shapes allowed by forests and short cycles.
Scientific publication4 min read
A finite-field proof establishes the product-size bound in a large dimension-gap regime and shows that equality requires one shared line across every family.
Scientific publication4 min read
A mathematical preprint describes a new family of permutation group polynomials over finite fields of odd characteristic. It pairs the family with an explicit companion, distinguishes it from known families considered in the paper, and gives exact counts for the new and three specified earlier families.
Scientific publication3 min read
An exact preprint finds that A4 and B4 admit a special vertex, while D4 and F4 do not. Three intermediate arrangements in the B4 deletion chain also pass the test, while the defect results remain limited to the named cases.
Scientific publication5 min read
A matching-complement theorem sits at the center of a deductive argument that establishes both the standard 2-Decomposition Conjecture and the 3-Decomposition Conjecture.
Scientific publication4 min read
A mathematical preprint establishes a sharp three-step cover for every 2-colored complete bipartite graph, while constructions show why stronger versions do not always hold.
Scientific publication3 min read
A theoretical study of rotation graphs gives a broad lower bound for the second-largest adjacency eigenvalue, reports fading spectral gaps in two graph families, and locates several stellohedron eigenvalues.
Scientific publication4 min read
A preprint proves Frankl's union-closed sets conjecture for finite families with inclusion-poset height at most four, and narrows possible height-five counterexamples to tightly constrained configurations.
Scientific publication4 min read
A mathematical preprint gives an asymptotically exact answer to a graph puzzle: 2n−4 edges are needed to make four shared edges unavoidable under every relabelling, for sufficiently large n.
Scientific publication3 min read
An arXiv preprint reports that two clique decomposition and cover conjectures hold for sufficiently large finite graphs, with equality tied to a named graph class in one problem and the Turán graph in the other.
Scientific publication4 min read
A new arXiv preprint proves a sharp minimum-semidegree threshold for deleting a directed path while preserving strong connectivity.
Scientific publication4 min read
A mathematical preprint links a graph’s tree-dimension to its crossing-graph coloring number, yielding a sharper cop-number bound for finite median graphs and exact formulas for several graph families.
Scientific publication4 min read
A mathematical preprint presents a closed formula for the dimension of the best-worst choice polytope and places it in the context of a reported four-alternative result.
Scientific publication3 min read
A theoretical study builds one algebraic language for stable series, differential operators and Jack functions, then tests selected low-degree formulas against three Gaussian matrix models.
Scientific publication4 min read
An arXiv preprint derives general bounds and selected exact formulas for isolation numbers across several graph products, while leaving a hypercube equality open.
Scientific publication4 min read
A theoretical paper finds the leading asymptotic minimum of clique-saturating non-edges and solves the triangle-free case exactly.
Scientific publication3 min read
A mathematical study of finite undirected graphs finds that n-convex geometries contain a star exactly when their number of vertices is odd, while a complete construction method for even-order cases remains open.
Scientific publication4 min read
A theoretical combinatorics study examines valid triangulations of cyclically colored convex polygons, finding exceptional three-color twist graphs and connected flip graphs for four or more colors.
Scientific publication3 min read
A mathematical preprint proves the signed list edge-colouring bound Δ + 1 for finite simple graphs of treewidth 3, and for treewidth 4 when maximum degree is at least 10.
Scientific publication4 min read
A mathematical preprint gives an exact classification of normal Ramanujan Cayley graphs in ratio-one Frobenius groups and counts the corresponding types for AGL(1,q).
Scientific publication3 min read
A mathematical preprint gives recursive formulas for binary-tree covering numbers and an exact structural result for survival covering on non-binary trees. It also shows that one unrestricted sequence can reduce every non-trivial subtree, while the ordinary covering number for general non-binary trees remains unresolved.
Scientific publication3 min read
A version 1 arXiv preprint claims that every rational exponent α ≥ 1 can describe the asymptotic count of one fixed graph inside graphs avoiding another, with the counted graph connected and of diameter at most 3.
Scientific publication4 min read
A combinatorics preprint uses slices, accessibility and marked vertices to derive exact formulas for several classes of plane hypermaps, including the one- and two-alternating cases.
Scientific publication5 min read
The study uses spectral and character-based calculations to separate graph constructions where proper fractional revival is impossible from those where exact algebraic conditions allow it.
Scientific publication4 min read
A mathematical preprint reports that the difference in facet counts between chain and order polytopes is exactly the sum of local star-element weights. It also classifies gaps of zero, one and two, and derives a corresponding formula for admissible decompositions in a regular marked setting.
Scientific publication4 min read
A mathematics preprint develops exact identities connecting partial permutations, class functions and forgotten symmetric functions. Its formulas lead to ribbon-tiling and rim-hook descriptions of irreducible-character evaluations, while a cancellation argument narrows one set-mapping calculation to partitions with at most two parts.
Scientific publication5 min read
A mathematical preprint reports a theorem showing that a positive gap below the three-quarter minimum-degree threshold is enough to rule out nonsynchronized local minima in the Kuramoto energy landscape. The result applies to finite simple graphs and does not specify the size of the gap.
Scientific publication4 min read
A mathematical preprint establishes the existence of a degree threshold guaranteeing vertex-disjoint directed cycles with pairwise different lengths. It also gives weighted extensions while leaving the sharpest possible threshold open.
Scientific publication5 min read
An arXiv preprint reports a universal bound between two graph-theory quantities, presents a constructive approximation algorithm, and gives examples showing why the factor two may be impossible to improve.
Scientific publication4 min read
A theoretical graph-theory preprint identifies an exact finiteness dichotomy for one family of forbidden-induced-subgraph classes, then supplies related finite, infinite and algorithmic results.
Scientific publication4 min read
A theory paper presents a canonical block-triangular decomposition for finite, connected, simple bipartite graphs. Its results connect the characteristic m to matching extension, structural classes and a lattice of independent sets, while leaving practical runtime performance untested.
Scientific publication5 min read
A deterministic graph-theory result identifies the exact minimum semidegree for prescribed directed 3q-cycles in the stated range, while leaving the best size cutoff unresolved.
Scientific publication3 min read
An arXiv preprint classifies four exceptional Deza graph constructions arising from tangent, secant and external-line relations on quadrics. It gives their exact parameters and structural descriptions, but leaves the full classification of relation unions in higher odd dimensions open.
Scientific publication4 min read
An arXiv proof note derives two deterministic upper bounds for nonempty, regular, increasing, 3-wise intersecting families of subsets. The sharper form uses the Lambert W function; a simpler Fourier argument gives a weaker bound.
Scientific publication4 min read