Problems

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.

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.

(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.

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…

Is it possible to partition K_9^3 into stars S_4 so that their mates partition K_9^4? (Star partition without the mate condition is possible [5].)

Does lim_n → ∞ hatr_∞(G_n) = 0 hold for every sequence of graphs (G_n) such that |V(G_n)| → ∞ and Δ(G_n) is bounded as n → ∞? What sequences (G_n) yield lim_n → ∞ hatr_∞(G_n) = 1?

Are there some graphs with diam(G)=n and diam(D_{2}(G))=⌈(1/2)diam(G)⌉+1.

For any d and any connected simple graph G of order d, is i(P_G,m) always a stable polynomial?

The radius of G_n is equal to n-σ(n)-1 .

Conjecture 6.2. Fix k,n \in N . If \leq is standard and x_{k-1}x_{l+1}=x_{k}x_{l} for all 0<k \leq l, then the sequence (e_{k-j}(n+j))_{j\geq 0} is PF with respect to \leq .

Given a non-negative number σ, is there a graph G with diameter 2, degree Δ and Δ^2+1-σ vertices?

We discuss ... including some new ones that we present in this last section (in particular Conjectures 11.3 and 11.5).