Problems

To replace this conjecture, a new conjecture is formulated that we coin as the Brualdi-Li tally conjecture.

Let G' be a graph obtained from a (Δ,1) -bidegreed graph G by inserting a bouquet in an edge of a bouquet internal path. Then (i) if ρ(G)<1+√Δ-1, then ρ(G^′)>ρ(G); (ii) if ρ(G)>1+√Δ-1, then ρ(G^′)<ρ(G); (iii) if ρ(G)=1+√Δ-1, then…

Does Theorem 1 hold with no restriction on K? If not, what is the least information needed on K?

it is not known whether \alpha_{\lambda}(k;p) has unimodal coefficients for all \lambda and k.

The element tildew_b is maximal in the weak order on tildeW/W among all dominant elements tildew ∈ tildeW/W : tildew^-1(0) ∈ S(b).

Let Y and Z be t-cross-intersecting sets in G_n whose sizes meet the bound in Theorem 1.2. If n is sufficiently large compared to t, then Y = Z and Y or Y^T is a t-coset.

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,…