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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?

It is conjectured that if division and reciprocation, as well as subtraction and negation, are also made allowed operations, then they are never necessary to achieve a maximizing combination.

A graph is irreducible by Y-Δ moves, pendant removal, self-edge removal, parallel reductions, series reductions, antenna jumping, and antenna absorption if and only if it has three medial strands which pairwise intersect twice, there is a…

Conjecture 2. f_(n,n,3),3=q^n+8[ cn+2 1 ]_q[ ln 3 ]_q[ l2 1 ]_q f_(n,4,4),3=q^14[ cn-2 2 ]_q[ ln 1 ]_q[ l6 1 ]_q-q^17frac(1-q^4)(1-q^n-3)^2(1-q^n-2)(1-q)^2(1-q^2)^2

Let G1 and G2 be two graphs that are P4-free and 2K2-free. Then the union of G1 and G2 is perfectly orderable.

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?

In particular, is it true that if the realization |\Gamma| of \Gamma through its direct complex \Delta(\Gamma) is a manifold, then the realization of its partial dual |\Gamma^S| is also a manifold?

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…

This raises the question of whether asymptotic separation between n and pdeg(f) is a strictly decreasing function when plotted against sensitivity order.

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

Conjecture. If G ∈ G(k,n) and if p= lfloorn/2⌋ orlceiln/2⌉ ifn ≡ 1(bmod 4),k iseven;n ≡ 3(bmod 4),k iseven, lfloorn/2⌋ ifn ≡ 1(bmod 4),k isodd, n-2/2orn+2/2 ifn ≡ 2(bmod 4),k iseven, lceiln/2⌉ ifn ≡ 3(bmod 4),k isodd, then R^′(G)≥…

Characterize König-Egerváry graphs, where varrho_e(G) = m(G) implies core(G) = ker(G).