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Let 2≤ℓ<k and n ≠5. If G is an n-vertex k-chromatic ℓ -connected graph,then i(G)≤ i(G^*).

If A and B are finite sets of distinct vectors in R^n then show that the order of the set D(A, B) = d(a, b) : a ∈ A, b ∈ B is at least min|A|, |B|.

In addition, we conjecture that also (⌈n/2⌉ − k)-RCS is in P for any constant k.

(2) χ(G(2,8,6))=4?

(i) If T is a p-tree, then θ(T) is single-valued if p ≥ 6. (ii) If T is a p-tree, then for each p ≥ 7: θ(T)= llp-2 ifΔ(T)geqslant p-3, p-1 otherwise. .

For each D ∈ D and ψ=ψ_D , there is a unique prime factor Z_ψ(x) of P_ψ(x) such that degZ_ψ equals the number of elements in G ψ .

For a given endofunction T, what is σ(d;T) ?

What is H(n;a_1,a_2,...,a_k) ? Is the upper bound given in Theorem 16 tight?

Conjecture 1 holds if X is assumed to be a compact metric space.

Every k-ary tangram T satisfies cut(T, S_k) ≤ c_k, for some finite constant c_k depending only on k.

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 .

If A is [k]^(4) then this set is precisely A itself, but is it always the case that (for |A| > 1) we have |S(A)| ≥ |A|?

If 2 ≤ c ≤ d, then β_2(K_d^2× K_c^2)=d(c-1).

Given a p × q integer matrix M with p ≥ 2, if none of the differences between two rows of M is parallel to 1^{T} , then m(M,n)=(2+o(1))n/log_{p}n.

Suppose now that b ≤ a < s, and m ≫ n^1+s-1. Then satex(n, K_1,s : m, K_a,b) = (1 + o(1)) minN(K_a,b, K_q^*), N(K_a,b, overlineK_r^*), where q = mint ∈ Z : N(K_1,s, K_t) ≥ m and r = mint ∈ Z : N(K_1,s, overlineK_t) > m.

(Brill-Noether Existence for ℝ-Divisors on Graphs) Let ρ(g,r,d)=g-(r+1)(g-d+r). Fix two real numbers r ≥0, 2 g-2 ≥d. If ρ(g,r,d)≥0 then there exists an ℝ-divisor of degree at most d and rank equal to r on G.

The family of graphs G_4,0,r, where r ≠ c^3 + 4c^2 + 1 + q and c ∈ N, excluding the graphs G_4,0,4, G_4,0,9, G_4,0,10 and G_4,0,17, is a family of Galois equivalent graphs with each P(G_4,0,r, λ) having Galois group S_4.

Conjecture 3.7. Let k be a positive 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-Y is (k-r(Y))-edge connected for all Y \subseteq V with r(Y) ≤k-2, (b)…

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

It is an important problem to determine if equality holds, since it would give an intrinsic description of the amplituhedron which does not mention Gr_k,n^≥ 0.