An open workspace for mathematical discovery. Check proofs, make comments, form collaborations.

Characterize the nontrivial lexicographic product graphs that are well-totally-dominated.

Is there a 2-factor in B_3 in which all cycles have length of at most 10?

We conjecture that this ring is Z⟨c,d,fracc^2k+1-e^2k+12⟩.

Let G be a bipartite digraph with sides A and B. For a set X ⊆ A let N^↔(X) denote the set of vertices, which have an edge to and from X. If |N^↔(X)| ≥ |X| - 1 holds for every X ⊆ A, then there exists a directed path that covers A.

For any n ≥3, the polynomial (x+1)A_n-1(x)+kx A_n-2(x) is Hurwitz stable if and only if k>-2E_n/E_n-1 .

lim_l → ∞ limsup_n → ∞ P_n^av(τ)(A_l;k_n^(n)) = 0, for all τ ∈ ∪_m=2^∞ S_m and for all k_n.

they conjectured that C_m × P_n is SEAT if m ≥ 4 even, n ≥ 3 and d ∈ 0, 2;

If G is a graph with maximum degree Δ(G) and k ≥ 2, then la_k(G) ≤ ⌈ Δ(G) · |V(G)|/2 ⌊ k|V(G)|/k+1 ⌋ ⌉, when Δ(G) = |V(G)| - 1, ⌈ Δ(G) · |V(G)| + 1/2 ⌊ k|V(G)|/k+1 ⌋ ⌉, when Δ(G) < |V(G)| - 1.

Araujo, Piga and Schacht asked whether density and codegree both above 1/4 force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is p_0 = max_0 ≤ x ≤ 1minx^3, 1-x ≈ 0.3177, and below it there are dense 3-graphs…

Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is Var⟨ MX, X⟩ ≤ C E|∇⟨ MX, X⟩|^2 for every symmetric M?

Han and Jiang asked whether being of klt type is an open condition in flat families of varieties. It is not.

Swinnerton-Dyer (1981) proved R-equivalence trivial on smooth cubic surfaces over p-adic fields with good reduction, except for three special types. The paper resolves two long-standing exceptional cases: triviality for the diagonal cubic…

For a transcendental entire function, how fast can |f(z)| be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus M(r, f)?

Given planks of fixed total width, how should they be placed to cover as much of a convex body's volume as possible? Karoly Bezdek asked whether, for a Euclidean ball, the optimum is a single plank centred at the origin. It is, and the…

A cyclic meander induces a cyclic permutation on its 2n marked intersection points. Schwartz's conjecture on the quadratic growth of the associated meander number is resolved.

Can there be a finite covering system of the integers with distinct moduli, all of which are odd and greater than 1?

For an inclusion-free hypergraph on n vertices, a weight assignment w:[n]→[d] is isolating when a unique edge attains minimum weight. Faber and Harris conjectured that the number of isolating assignments is at least n∑_j=0^d-1 j^n-1,…

The low-degree conjecture predicts that when the low-degree advantage between a planted distribution and a uniform null distribution stays bounded, no polynomial-time algorithm can distinguish them. It is false. There is a planted…

Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. Their conjectured lower bound is established for…

Norine conjectured that every red-blue edge-colouring of the n-dimensional hypercube Q_n in which antipodal edges get opposite colours contains a monochromatic path from some vertex to its antipode. The paper proves it, via a chain-level…