Problems

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

While it is important to note that twisted subgroups need not be subgroups (e.g., there are small counterexamples in non-abelian groups of order 27 and 75), it could perhaps be the case that L(G) = {|H| : H \subsetneq G} (and thus, our…

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).

The characteristic set of a path-star tree contains an edge.

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) .

Given a finite CW complex X which is not contractible and two ρ-immersed matroids (M, l) and (N, l') such that T_X(M, l) ≃ T_X(N, l'), if there exists a surjective weak map τ: M → N, then τ^# is an isomorphism.

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}) .

A minimal forbidden induced subgraph for the property c_2(G)≤ k has at most 2 k+2 vertices.

• ρ^⊥(G) + ρ^⊥(barG) ≥ |V(G)| - 2 • ρ^⊥(G) + ρ^⊥(barG) ≤ |V(G)| + 2

For any triple of positive integers a = (a_1, a_2, a_3) the sequence of numbers the sequence g_i(a)_i=0^∞ is monotonically increasing with i, for i ≤ 14.

Identify and characterise the product graphs whose curling numbers are the product of the curling numbers of their factors graphs.

It is possible to adapt Theorem 1 to arbitrary posets (instead of 0, 1-posets) and to injective maps φ with x < y ⇔ φ(x) < φ(y) (instead of x ≤ y ⇔ φ(x) ≤ φ(y))?

We have max_t m_H_t(2, ∞) = max_t m_G_t(2, ∞) and max_t m_H_t(-∞, -2) = max_t m_G_t(-∞, -2).

We believe that results similar to Theorem 4.3.5 can be proved for intersection graphs of other scalable objects. In particular, we conjecture that similar techniques apply to intersection graphs of (unit) regular hexagons.

For k,n,α ∈N , let G ∈ G_n,α have the minimum spectral radius in G_n,α . Then for sufficiently large n, (1) G ≅ F(k,k,k+1) for α=3,n=3k+1, (2) G ≅ F(k+1,k,k+1) for α=3 and n=3k+2, (3) G ≅ F(k,k,k,k+1) for α=4 and n=4k+1, (4) G ≅…

Characterize non-König-Egerváry graphs satisfying; • varrho_e(G) ≥ m(G) - ξ(G) + ε(G); • varrho_v(G) = n(G) - ξ(G) + ε(G); • varrho_v(G) = α(G) + μ(G) - ξ(G) + ε(G).

Characterize the (m, n)-extremal graphs for all m and n.

We believe the assertion of Theorem 1.6 is true even for all smaller values of n, though we don't have a proof yet.