Problems

If F = F_q is a finite field of odd size or F = F_∞ is an algebraically closed field of characteristic zero, then every graph Γ_F(f_2, f_3) of girth at least eight is isomorphic to Γ_3(F) = Γ_F(xy, x^2y).

It would be very interesting to prove the full version of Theorem 4.1 in a similar manner.

Is it true, for example, that all values S(d, k) are even? In other words: If an odd subset of E_d is sliced by k hyperplanes, can one always add another edge to this set?

Let D be a k-arc-connected digraph and let l ≤ k. If (x_1, f_1, y_1), …, (x_l, f_l, y_l) are l triples such that x_1, …, x_l, y_1, …, y_l ∈ V(D) (not necessarily distinct) and f_i ∈ E^+(x_i) (respectively f_i ∈ E^-(y_i)), i = 1, …, l, then…

Suppose that k>ℓ . Then satex(n,P_k:m,P_ℓ) is attained asymptotically on the quasi-star or the quasi-clique.

How closely related are σ^2 and λ ? In particular, is it true that λ=Θ(σ^2) (that is, are there bounds on the ratios λ/σ^2 and σ^2/λ )?

If G is a subcubic planar graph drawn without any faces of length 5, then χ(G^2)≤ 6.

It is however conjectured that one of these constructions either “K-groupings” or “J-groupings” will yield a maximum independent set for a given I-graph.

Conjecture 21 states that the equality between the summations in Theorems 19 and 20 holds term-by-term.

There are no regular self 2-distance graphs of odd degree.

It would be interesting to see if a certain multivariate generating polynomial of Mahonian–Eulerian statistics, such as ∑{π∈𝔖_n} q^maj(π) ∏{i∈𝒟(π)} x_{π_i}, is stable.

For any n, r ∈ N with r ≥ 3 and n ≥ (r-1)^2 + 1, the set MUC(n, r) is not empty.

Again, it is open whether a similar result holds for x-monotone or radial drawings.

For each n, there exists a term order such that a code is 0- or 1-inductively pierced if and only if the reduced Gröbner basis contains binomials of degree 2 or less.

(Analog of Samotij's theorem in Z_2^n). Let n ≥ k ≥ 2 and M be integers. Amongst all families F ⊆ Z_2^n of size |F| = M, centred families minimise the number of 2^k-cubes.

Given a (2s + 1, k, λ) difference set D in a group G, and given a graph Γ of order k and size s, we ask whether it is possibile to label the vertices of Γ with the elements of D is such a way that every non-identity element of G may be…

For any n, p ∈ N with n ≥ p+1, G = P_n^p is equitably k-list arborable if and only if k ≥ ⌈ (p+1)/2 ⌉.

We believe c(Q_n)≤ c(Q_n+1) (and similarly for c_L ), but a proof has eluded us.

Is C_4 in DP? More generally, are even cycles in DP?

Let w ∈ S_n be a permutation and l := ℓ(w) be its length. Denote by CS(w) = a = (a_1 ≤ a_2 ≤ … ≤ a_l) ∈ N^l the set of compatible sequences [7] corresponding to permutation w. Define statistics r(a) on the set of all compatible sequences…