Problems

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

Assume that p nmid (m-1) and 2 ≤ m ≤ q+1/3. Then the cliques from Proposition 4.10 and Proposition 4.13 are maximal.

Any collection of sets with no empty Venn regions is splittable.

If given two positive integers a and b where a ≤ b ≤ 2a, is it possible to find a graph and a permutation α on V(G) such that γ(G)=a and γ(P_α(G))=b for all a,b ∈ N?

All perfect strategies of G(G,b) have classical correlation matrices if and only if b has even parity.

What is the S_n -module structure of tildeH_3k-4(NM_k(n)) ?

If L is a set of lines in F_q^3 such that |L|=Ω(q^3) , and such that no plane contains ω(q) lines of L, then |P(L)|≥(1-o(1))q^3 .

The sequence of coefficients of F_n^(a)(x) is unimodal.

Find a derangement σ of the set M of all noncrossing perfect matchings of 1, 2, …, 2n such that ⟨ α, σ(α) ⟩ = 1 for all α ∈ M.

Do there exist polynomials f_{1}(\zeta),f_{2}(\zeta),f_{3}(\zeta) and f_{4}(\zeta) ,corresponding to every pair (a, b), satisfying (33), given as in (15), satisfying (16)and (17), such that f_{1}(1)=0,\quad f_{2}(1)^{2}=a^{2},\quad…

Let G be a finite graph and G = G(min). What condition on the set \mathcal{K}(G) = {K(v) | v \in V(G)} is equivalent to the existence of an endomorphism of G' modulo G?

Is M(u) a CFG or not?

Is this true? “For each positive integer n, there is a graph G with n vertices such that dis[G] - dis_s[G] = Ω(√n).”