Problems

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 ?

It remains open whether enumeration on chordal graphs can be improved further, so we hereby pose it as an open problem, or whether one can obtain a higher lower bound, which might also be a gap-improvement on general graphs.

Conjecture 18. The top subsegment of the n-core (n=12,14,16, ...) is μ_n/4=μ_n-4(A-3× 2^n-7)⊕ μ_(n-2)/4(A)^3 , where A=M_n-1+2^n-3 is the senior term of the n-core.

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

Suppose we revised our definition of an s-partition so that each part was required to be 2-edge connected, except the small part. What degree of edge connectivity would be required to ensure the existence of an s-partition (if such a…

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

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).

Moreover, we conjecture that the only finite singularities of Φ_q(t) are of the form q^m/(q-1), m ≥ 1.

Let G be a graph with order n and size m. Then λ_n(A_1/2(G))≥ m/n-1-n-2/2.

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?

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?

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