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For m ≥ 2, the sequence d_i+1(m)d_i-1(m)/d_i(m)^2_2 ≤ i ≤ m-2 is reverse ultra log-concave.

There is a constant C such that every connected infinite planar graph with subexponential growth contains a one-way-infinite path P=v_1v_2v_3... such that for every k ≥ 1 ∑_i=1^kdeg_G(v_i)≤ Ck log k.

We close this paper with the following conjecture: all r-dimensional grids, with finitely many exceptions, are domatically full.

• ρ^⊥(G) + ρ^⊥(barG) ≥ |V(G)| - 2 • ρ^⊥(G) + ρ^⊥(barG) ≤ |V(G)| + 2

As a first step, we conjecture that if a graph G has no extreme vertices and |∂(G)|=4 , then ∂(G)=Per(G) .

There exists a bijective map that maps each perfect matching of a 2-by-2-by-2n cube snake to an ordered pair of perfect matchings of the 3-by-2n grid. Additionally, there exists a bijective map that maps half of the perfect matchings of a…

Given a signed graph consisting of two identical cliques con nected by a single edge S=((K_n∪ K_n)+e,σ) , show that χ'(S)=Δ(S)=n .

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?

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 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,…