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Can Theorem 4.2 be true for dimension ≥ 4 ?

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).

The complement of multicone graphs K_w ∇ L(P) are DS with respect to their signless Laplacian spectrum.

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

Can we find the limit?

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

Let G ∈ G_3(n,n-3) be a graph of order n ≥ 6. Then the following cases hold: i) if β<α<0<γ<ρ , then G is Seidel equivalent to K_i,j∪ barK_p ; ii) if ρ<γ<0<α<β , then G is Seidel equivalent to overlineK_i,j∪ barK_p ,where 1≤ i ≤[n/3], i≤ j≤…

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?

By (10) they occur in inverse pairs, with 1 an eigenvalue for all odd n; how big is the largest?

The +1 in (3) can be replaced by +1 / 2 (which is best possible).

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) .

Let (5.3) δ_o(j) = 0 if j is even 1 if j is odd. Define r_k(n) by (5.4) r_k(0) = k r_k(j) = 2^j-δ_o(j)· 2, for 1 ≤ j ≤ k-1 r_k(n) = 2 ∑_l=0^k-2r_k(n-(l+2)), for n ≥ k. Then, S(R_2,3,...,k(n))=r_k(n) for all values of n ≥k.

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?