Problems

Let G be a connected non-transmission-regular graph with n vertices. Then D_1 - λ_1(D) > 1/n+1.

Let K_n_1,n_2,...,n_r be a complete r-partite graph with r vertex sets X_i(i ∈[1,r]) and |X_i|=n_i , ∑_i=1^r=n . Besides (i) and (ii) in Theorem 8, fgndi_∑(K_n_1,n_2,...,n_r)≤ 3 ?

What are the homotopy types of Q_n(Π) ? (in general their order complexes aren't necessarily spheres, or even Cohen-Macaulay)

Another problem worth mentioning is whether the lower bound for c_2(G) still holds without the regularity assumptions, i.e. if we only assume that the graph has large girth and the degree of each vertex is greater than 2.

Let X be a compact Hausdorff space and T:X → X a continuous map. For any open U ⊆ X and any ℓ ∈N , there exists n ∈N with U ∩ T^-nU ∩ T^-2nU ∩… ∩ T^-ℓ nU ≠∅,or (3) T^-inU ∩ T^-jnU=∅ ∀0 ≤ i<j ≤ℓ. (4)

Is this true “for any graph G, dis_s[G] ≤ χ(G)?”

For any nonnegative integer k, does there exist a connected graph G satisfying φ_(2,j)(G) − κ_(2,j)(G) + 1 = k?

Does a Nim-basis, if it exists, necessarily consist of the disjoint unions of circuits of the complex?

Is there a graph G such that for any non-empty graph H, we have χ_{td}(H □ G) > χ_{td}(G)?

There is a constant K such that for every 2-edge-connected plane graph G it holds fep(G) ≤ K.

Do we always have ∂_v,G_B(A)≥ d μ(C(B))^1/d|A|^1-1/d ?

For any x ∈(-3,c) with x ≠ 0, we have ∑_k=0^∞fracx^3k(k+1)(x-1)^k( c3k k )= frac3(x-1)2x^3log^2(1-x)+frac2(1-x)x^3q(x)^2 +s(x)log(1-x)+fract(x)q(x)√(1-x)(3+x), (1.17) where s(x) and t(x) are suitable rational functions in x.

Given r random vertices v_1,...,v_r of C^d , what is the expected number of 0 / 1-vectors in the affine subspace spanned by these vectors?

Maximal in size concept lattice of a formal context (G, M, I) of VC-dimension at most k, such that |G| + |M| = 2n, and such that k divides n, is the Cartesian product of k chains of length n/k - 1 each: L = bigotimes_k C(n/k), where C(l)…

If D is a dimatroid and C,D ∈ D , then there exist C',D'∈ D of almost equal size whose union is C ∪ D .

What is limsup_n →∞frace(G)binomnr:G ⊆( c[n] r ),G containsnosubgraphthatcoverspairs?

Let ε > 0 be any constant and let q be a sufficiently large prime power. Let L be a set of at least q^5/2+ε lines in F_q^3 such that no plane contains more than (1/2)q^3/2 lines of L. Then, |P(L)| ≥ (1-o(1))q^3.

Is it true that f_DR^2(n,C)≈3/2n ? If not, can one prove at least, that f_DR^2(n,C)<(2-c)n? (34)

Given k, does every circle in an edge-minimal k-highly connected standard subspace X of |G| contain a vertex or end whose degree in X is at most k ?

For each vertex v in a graph G of order n ≥ 4, b(G-v)=b(G)+lfloorn/2⌋-2 if and only if G=C_4,P_4,2P_2