Problems

Open problem: • d∈0,2 for n>7

Pascal-type behavior except for the entry 14.

If we remove the equal coefficients condition, does Theorem 4.1 hold with n = d + 1?

Is there some C(r)>2 such that ρ_2(2(r+2)+1,r+2)≥C(r)ρ_2(2r+1,r) holds for all r ≥2 ?

Suppose G is a Hamiltonian chordal graph. Is G cycle extendable if min{Δ(T) : (T, T) is a tree decomposition for G} = 4?

It seems an interesting problem to characterize semisymmetric graphs with Wiener dimension 2.

Could it be true that for any finite dimensional \mathfrak{gl}{n} -module W there exists a polynomial p{W}(t)(p_{W}(t)=t ??) such that for all partitions \pi and \mu one has if a_{N \mu,W}^{N \pi}\ge p_{W}(N) , then a_{\mu,W}^{\pi}\ne 0 .

Characterize the graphs G such that every irreducible dominating set in G is either a minimal dominating set or a minimal total dominating set.

CONNECTED TREEWIDTH can be solved by an O(n^f(k))-time algorithm.

Let d ∈ [0, n] and E ⊆ [0, n]. For j ∈ [0, n], if j ∉ Z-cl_n,d(E), then there exists a polynomial P(X) = ℓ_1(X) … ℓ_k(X)σ(X) ∈ F_p[X], where deg ℓ_1 = … = deg ℓ_k = 1, and σ(X) is a symmetric polynomial, such that deg P ≤ d, P|_underlinei…

It remains unknown to us whether or not there exists a divisional but not inductive poset among non-lattice, locally geometric posets.

For positive integers k, d and n with k ≥3, find the largest value f_{k,d}(n) such that every connected graph G of maximum degree at most d and of order n contains a k-tree T with |T|≥f_{k,d}(n).

If G is a critical graph (not banned) and ρ^⊥(G)=d , then G-M(G) (with M a match maximal) has ρ^⊥(G-M(G))≤ d-1

Can one prove better upper or lower bounds for ℓ_n ?

The question of finding necessary and sufficient conditions for this to happen is to the best of the author's knowledge an open problem.

Under which conditions a)d(G_1,G_2)=q_1+q_2+|p_1-p_2|-2,b)d(G_1,G_2)=q_1+q_2+|p_1-p_2|-4 hold?

If F = F_q is a finite field of odd size or F = F_∞ is an algebraically closed field of characteristic zero, then every graph Γ_F(f_2, f_3) of girth at least eight is isomorphic to Γ_3(F) = Γ_F(xy, x^2y).

Can we prove that they are actually convergent to the same limit?

The maximum possible load, on any vertex in any graph, is 1/8n^3-O(n^2) .

But we conjecture that at least one of π and π^{-1} will always have a sufficiently large strong compatible set to ensure a better approximation for bs(π)=bs(π^{-1}) .