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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).

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.

Open problem: • d ∈ 3, 4 for n ≥ 1

Conjecture 5.1. For β>0 , the only 2-ary, order 2(α,β) -Conolly recurrences are

The Möbius function of KC_n alternates in sign for each n ≥ 1.

Let χ(H)≤ k≥ 3 . Then rex(n,H,K_k+1)=(1+o(1))N(H,T(n,k)) .Moreover, if n is sufficiently large and is divisible by k, then rex(n,H,K_k+1)=N(H,T(n,k)) .

Determine the complexity of INDEPENDENT DOMINATION OF DIRECT PRODUCTS