Problems

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.

For every K_r -free graph G, fracα(G)overlineα_G(1)≥ 1+1/r. For every K_r -free graph G of minimum degree d, fracα(G)overlineα_G(1)≥ 2-o_d(1), with r fixed as d →∞ .

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.

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).

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?

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 \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,…

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…

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