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But we conjecture that at least one of π and π^{-1} will always have a sufficiently large strong compatible set to ensure a better approximation for bs(π)=bs(π^{-1}) .

The radius of G_n is equal to n-σ(n)-1 .

A minimal forbidden induced subgraph for the property c_2(G)≤ k has at most 2 k+2 vertices.

Conjecture 6.2. Fix k,n \in N . If \leq is standard and x_{k-1}x_{l+1}=x_{k}x_{l} for all 0<k \leq l, then the sequence (e_{k-j}(n+j))_{j\geq 0} is PF with respect to \leq .

For every graph G = (V + s, E) with d(s) ≥ 4 there exist rs, st ∈ E such that for every best-balanced orientation vecG_rt of G_rt, vecG := vecG_rt - rt + rs + st is a best-balanced orientation of G.

Given a non-negative number σ, is there a graph G with diameter 2, degree Δ and Δ^2+1-σ vertices?

We discuss ... including some new ones that we present in this last section (in particular Conjectures 11.3 and 11.5).

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 .