Problems

For an arbitrary planar graph G, is there a proper grid drawing of G in a grid of polynomial size?

For each D ∈ D and ψ=ψ_D , there is a unique prime factor Z_ψ(x) of P_ψ(x) such that degZ_ψ equals the number of elements in G ψ .

Conjecture 1 holds if X is assumed to be a compact metric space.

Given a p × q integer matrix M with p ≥ 2, if none of the differences between two rows of M is parallel to 1^{T} , then m(M,n)=(2+o(1))n/log_{p}n.

(Brill-Noether Existence for ℝ-Divisors on Graphs) Let ρ(g,r,d)=g-(r+1)(g-d+r). Fix two real numbers r ≥0, 2 g-2 ≥d. If ρ(g,r,d)≥0 then there exists an ℝ-divisor of degree at most d and rank equal to r on G.

In PG(3, q) and PG(4, q), the upper bounds (1.5), (1.6) hold for all q.

(Covering Radius Conjecture) Let λ ∈ [1/g, g] (recall that g ≥ 1). The covering radius of N_G with respect to the polytope P_{1,λ} is at least √(g/λ)/n where n is the number of vertices of G.

Perhaps an equally daring conjecture would be that L(G) = {d : d divides |G|}, in which case we would have f(G) = f^{*}(|G|).

Conjecture 1. Given an undirected connected graph G. We consider v an extremum of the Fiedler vector of the graph G.\tilde{G} is the graph obtained from G and v as in Proposition 1. Then for all x>0 the Fiedler vector \Phi(x,\cdot) of…

Let M(n) denote the absolute value of the Möbius function μ[1,W_n]=μ[1,M_n] . Then for n>50 we have M(2n)=n^2⇔ n+1 isprimeandn ≡ 0(bmod6)M(2n)=n^2-1 ⇔ n+1 isprimeandn ≡ 4(bmod6)M(2n+1)=n^2-n ⇔ n+1 isprimeandn ≡ 0(bmod6)M(2n+1)=n^2-n-1 ⇔…

It remains open whether enumeration on chordal graphs can be improved further, so we hereby pose it as an open problem, or whether one can obtain a higher lower bound, which might also be a gap-improvement on general graphs.

Conjecture 18. The top subsegment of the n-core (n=12,14,16, ...) is μ_n/4=μ_n-4(A-3× 2^n-7)⊕ μ_(n-2)/4(A)^3 , where A=M_n-1+2^n-3 is the senior term of the n-core.

There exists a chain complex of finitely generated FI-modules C_* such that H_k(C_*) = H_k(K_p(S_•)).

Let f(x_1,…,x_n) be a polynomial over a field F given by (1.1) and (1.2). Provided n≥ k, for any finite subset A of F we have |(f(x_1,…,x_n):x_1,…,x_n∈ A, and x_i≠ x_j if i≠ j)| ≥ minp(F)-llbracket n=2 a_1=-a_2rrbracket,…

We conjecture that this is true for all q.

In which condition power graph P(G) of a non-degenerate gyrogroup G is complete?

Let V be a 2-dimensional subspace in F_p^4, such that 1 ∈ V. Then V is a clique in PP(p^4, p + 1, I) for some I if and only if V = F_p ⊕ aF_p, where a = g^(p+1)k and k is an odd integer.

If a neural code is labeled such that the i^th neuron is added as a piercing at the i^th step, then its toric ideal has a quadratic Gröbner basis with respect to the term order prec.

Specifically, we conjecture that similar processes will work for cycle pendant stars up to stars of size 15.

We conjecture that i_k(n)/i_k-1(n) is a decreasing function of k for any n.