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We conjecture that the number of tilings of any finite contiguous C by tiles of size α is an upper bound on the number of tilings of any finite C'⊂ Z^d by tiles of size α .

Find a function b(k) such that for k ≥ 3, the following bound is true and tight for connected graphs G: b(k) · γ_t(G) ≤ γ_krt(G).

If P, Q ⊂ R^d are any rational polytopes, then we have: σ_P(ξ^*) = σ_Q(ξ^*) ⇒ P = Q, with ξ^* as in (4).

independence_number(x)>= ceil(lovasz_theta(x))-girth(x)

For m,n with m≤ n, what is a good general lower bound for γ(Q_m× n)? In particular, is it true that γ(Q_m× n)≥ minm-1,⌈ n/2⌉-1?

In which condition power graph P(G) of a non-degenerate gyrogroup G is complete?

Let G be a finite transitive group on Ω. If I_Ω(G) > 1/2, then I_Ω(G) = (q+1)/2q, for some q ∈ Q with 2q ∈ N.

Is it possible to embed, a given unicyclic graph in a graceful unicyclic graph?

It would be of interest to determine whether the growth of \sup_{i\le n}NMC(i) is exponential as n \to\infty or if the subsequence of strictly increasing terms is exponential.

For any positive integer t, is there any bipartite graph G such that dis_ℓ[G] - dis[G] ≥ t?

Show that 2 ·1_k-1 and 2 ·0_k-1 are the only (k-1)-rowed critical sub structures of K_k .

Let V be a 2-dimensional subspace in F_p^4, such that 1 ∈ V. Then V is a clique in PP(p^4, p + 1, I) for some I if and only if V = F_p ⊕ aF_p, where a = g^(p+1)k and k is an odd integer.

Let b: V → Z_0^+ be a symmetric crossing submodular function with b(∅) = 0 and b(X) ≡ |X ∩ T_b| mod 2. Then there exists a pairing M on T_b that satisfies (17).

Does our improved estimate for the edge isoperimetric inequality in the remark following Theorem 3.1 hold for general B, i.e. do we always have ∂_e,G_B(A)≥ d μ(Z(B))^1/d|A|^1-1/d ?

Let \lambda be a partition, \mu and \eta be compositions such that |\lambda|=|\mu| and ll(\mu)\le |\eta| . Then the coefficient c(\lambda,\mu | \eta) is a homogeneous piecewise linear function of \lambda and \mu. In particular, c(N\lambda,…

For any triple of positive integers a = (a_1, a_2, a_3) the sequence of numbers the sequence g_i(a)_i=0^∞ is monotonically increasing with i, for i ≤ 14.

More generally, call a set of subgraphs of G a packing if the subgraphs are disjoint. Let f(G) be the number of packings of G by copies of a fixed graph K (so when K is an edge, this is the number of matchings). Does (5.3) hold when H…

Identify and characterise the product graphs whose curling numbers are the product of the curling numbers of their factors graphs.

There is no graph with uniform rank spread equal to two.

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