Problems

Is it true that for every nonnegative integer k, there exists a connected hypergraph H satisfying φ_ST(H) − κ_ST(H) + 1 = k?

Let G be a 2-connected bipartite graph with sides A and B satisfying |N^2(X)| ≥ |X| for every X ⊆ A of size at least 3. For every X ⊆ A, |X| ≥ 3, there is a cycle C_X in G such that V(C_X) ∩ A = X.

Let G be a fuzzy graph. Then, (1) s(G) ≤ 2Δ(G) + 1, (2) for every integer k ≥ 2, there exists a fuzzy graph G_k such that k-1 ≤ Δ(G_k) and s(G_k) = Δ(G_k) + k.

First of all, is it possible to sharpen Theorem 1 to the assertion that ‘colex is best’: if A ⊂ [n]^(3), and C is the set of the first |A| elements of [n]^(3) in the colex order, then must we have |VA| ≥ |VC|?

Suppose that G ∈ G^r . Is λ_min^(p)(G) continuously differentiable for p>r ? Is λ_min^(p)(G) continuously differentiable for p \ne k, k=2, ..., r ?

For which graphs H does there exist an H' such that π_G^(H') determines π_G^(H) for every graph G?

Conjecture 3.6. Let k ≥2 be an integer, G=(V,E) be a graph, and r:V \to Z_{+} such that r(V) ≥k+1. Then G has a k-connected r-detachment if and only if (a) G is k-edge connected, (b) d(v) ≥k r(v) for all v \in V , (c) G-y has a…

We conjecture that ex_v(vecV_r, vecQ_n) = 2^n-1 + Θ(n^r-2) holds for every r ≥ 3.

Let f(n) be the largest integer for which there is a C_4 free graph of n vertices every vertex of which has degree ≥ f(n). Is it true that f(n+1) ≥ f(n)?

Let d ≥ 1. Then for any 0 ≤ ℓ < k ≤ d - ℓ - 1, does there exist an infinite family P_1, P_2, … of integral convex polytopes of dimension d such that for each P_i and P_j with i ≠ j, the followings are satisfied: For t = 1, …, k, we have…

If r = o(n) holds, then the order of magnitude of M(n, r) is Θ(n).

If M and M / e are both non-degenerate, then Q_M(t) interlaces Q_M/e(t).

Specifically, is there an absolute positive constant c so that any connected graph with minimum degree at least d contains a spanning tree in which the degree of any non-leaf is at least cd/log d?

If χ(G-v)<χ(G) then α_*(G-v)≤α_*(G) .

What about the number of real-valued zonal spherical functions?

Let G be a graph. If r is an integer root of D_t(G, x), then r ∈ -3, -2, -1, 0.

It would be very interesting to prove the full version of Theorem 4.1 in a similar manner.

Is it true, for example, that all values S(d, k) are even? In other words: If an odd subset of E_d is sliced by k hyperplanes, can one always add another edge to this set?

Let D be a k-arc-connected digraph and let l ≤ k. If (x_1, f_1, y_1), …, (x_l, f_l, y_l) are l triples such that x_1, …, x_l, y_1, …, y_l ∈ V(D) (not necessarily distinct) and f_i ∈ E^+(x_i) (respectively f_i ∈ E^-(y_i)), i = 1, …, l, then…

Suppose that k>ℓ . Then satex(n,P_k:m,P_ℓ) is attained asymptotically on the quasi-star or the quasi-clique.