Problems

We conjecture that the necessary conditions are sufficient in general, except eventually for a few values (for example it can be shown that K_{4,4,4,1} cannot be decomposed into K_4's).

W(n,k,4)=0 if either (i) n<14 or (ii) n=14,k<14.

Do either the height 1 or Hilbert basis extensions generate the same poset Cones(d)?

We conjecture that if a Γ -degree sequence d' has a tree realization then each such a realization of d' has the same number of pendant vertices.

If D is a digraph of order n, then d_I(D) + d_I(barD) ≤ n + 1.

For all primitive digraphs G such that G ≠ K_n^*, exp(G)/l(G) ≥ 2.

The construction described above has Ω(n^6) crossings. Does there exist a cycle of small area in every drawing of K_n such that every pair of edges intersect a constant number of times?

What is the smallest n for which Φ_n(G,H)= r(G, H)-2 ? Does this relation hold for all n ≥r(G, H) ?

For every weakly distance-regular digraph Γ with valency k, the edge connectivity equals to k. Moreover if k > 2, any minimum edge cut is the set of all edges going into (or coming out of) a single vertex.

The set E_n is a minimal generating set in the strong sense: E_n has the lowest cardinality of any generating set of C_n .

If G is a connected graph containing a percolating set which r-percolates in k rounds and r ≥2, then k≤ diam_D(G) .

The complete graph possesses the largest eigen-cover area of all classes of graphs.

Does it make a difference if for every vertex v on level i we also prescribe a connected component of {(x, y, z) ∈ R^3 | z = i} ∩ S to which v belongs in a drawing?

We noticed how similar these are to the asymptotics of the sequences enumerating 123-avoiding words with r occurrences of each letter, given on page 8 of [SZ], and we have a similar conjecture as on page 3 of [SZ] that a_r(n) is…

does there exist a packing k-colouring so that, for i<j ≤ k the asymp-totic frequency of colour i is no more than the asymptotic frequency of j?

Specify the size sequences of ⟨ℤ; -⟩ for each k ≥ 2.

(P2) If q is an integer such that γ(n, r) ≤ q ≤ η(n, r), can we find n real numbers a1, …, an, such that r of them are non-negative and the remaining n − r are negative with ∑_{i=1}^n a_i ≥ 0, such that the number of the non-negative sums…

Finally, in chapter 5, we give a conjecture that every graph with at least one edge has an effective competition cover.

For positive integers n_1 ≥ n_2 ≥ … ≥ n_t ≥ 2, we have R(P_2n_1, P_2n_2, …, P_2n_t) = n_1 + ∑_i=1^t (n_i - 1) + 1.

At the same time we suppose that the factor 3 in this condition is "conventional", i.e. we conjecture that for any natural k ≥ 3 after replacing this condition by p(r) ≥ kp Theorems 22-24 will remain true.