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Whether fgndi_∑(G)≤ 3 holds for every connected graph G with Δ=3 ?

A warmup problem (which I have no idea how to solve, but which is experimentally plausible) is to show that given any line segments L_1,...,L_m in R^n , the Steiner polynomial p(t_1,...,t_m)=Vol(t_1L_1+...+t_mL_m)is hyperbolic.

We conjecture that this is true for all q.

Is it true that all primitive tiling periods are minimal-size tiling periods?

We conjecture that SAGE(n, k)=n-1 for any n, and k>n.

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) .

Suppose 17/30≤α≤1 and β>(2 α-1)/4 or 43/120≤α<17/30 and β>(5 α-2)/25 . For all sufficiently large n, a graph of order n in which each set of ⌊α n⌋ vertices spans at least β n^2 edges must contain a triangle.

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?

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.