Problems

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.

How closely related are σ^2 and λ ? In particular, is it true that λ=Θ(σ^2) (that is, are there bounds on the ratios λ/σ^2 and σ^2/λ )?

If G is a subcubic planar graph drawn without any faces of length 5, then χ(G^2)≤ 6.

Conjecture 21 states that the equality between the summations in Theorems 19 and 20 holds term-by-term.

There are no regular self 2-distance graphs of odd degree.

It would be interesting to see if a certain multivariate generating polynomial of Mahonian–Eulerian statistics, such as ∑{π∈𝔖_n} q^maj(π) ∏{i∈𝒟(π)} x_{π_i}, is stable.

For any n, r ∈ N with r ≥ 3 and n ≥ (r-1)^2 + 1, the set MUC(n, r) is not empty.