Problems

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).

Whether fgndi_∑(G)≤ 3 holds for every connected graph G with Δ=3 ?

A warmup problem (which I have no idea how to solve, but which is experimentally plausible) is to show that given any line segments L_1,...,L_m in R^n , the Steiner polynomial p(t_1,...,t_m)=Vol(t_1L_1+...+t_mL_m)is hyperbolic.

Is it true that all primitive tiling periods are minimal-size tiling periods?

We conjecture that SAGE(n, k)=n-1 for any n, and k>n.

For m ≥ 2, the sequence d_i+1(m)d_i-1(m)/d_i(m)^2_2 ≤ i ≤ m-2 is reverse ultra log-concave.

There is a constant C such that every connected infinite planar graph with subexponential growth contains a one-way-infinite path P=v_1v_2v_3... such that for every k ≥ 1 ∑_i=1^kdeg_G(v_i)≤ Ck log k.