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For the case k>5, Corollary 4.19 gives a necessary condition for what kind of cycles can appear in the promotion action on SYT(sc_k) . We do not know if this condition is sufficient.

B(n, m) = n(m-1) + 1.

If G is a connected graph of order n ≥ 4 and ρ_ABC(G) ≤ √2, then G ∈ P_n, C_n, S_4.

Based upon the results generated from our Sage script, we submit as a conjecture that these graphs constructed be the smallest graphs (by order) that have characteristic-dependent well-covered dimension for any given characteristic.

Conjecture an explicit classification of unfolding trees of graphs with two vertices.

Use Theorem 6 to formulate and proof a generalized result for complete q-partite graphs.

There exist k,r ∈ \mathbb{N} and a pattern v ∈ [k]^{*} such that d>1 and K_{r} in (9) is equal to zero. In that case, there exists L_{r}∈ \mathbb{N},L_{r}<M_{r} , and \tilde{K}{r}∈(0,\infty) such that \lim{n…

Let δ≥ 3 be an integer. Does there exist a δ -chromatic quadruple system Q such that χ(K(Q))=δ ?

However, for the minimization problems, we do not know whether fractional treewidth-fragility is sufficient even for the distance-1 problems.

When G is a tree, is it true that the Hilbert stratification of A_G consists of coordinate subspaces? In particular, does f=e^u give a general Hilbert sequence?

Thus we can ask a question: "Can we develop an enumeration method to find the number of perfectly dominated trees of order n?"

the existence of such an f has been proved, but uniqueness in T_0 has not.

Our result leaves open the question about the largest cardinality of a set of pairwise completely K-different permutations.

Let G be a connected graph of order n. Then G is a star graph if and only if n_0(e;k)=n_u(e;k)=0,n_v(e;k)=( cn-2k-1 ) or n_0(e;k)=n_v(e;k)=0,n_u(e;k)=( cn-2k-1 ) for any edge e=uv ∈ E(G) .

Let G be a Hamiltonian bipartite graph of minimum degree δ on n vertices, where n<2(δ^{2}-δ+1) . Then G has a cycle of length 2 l for each integer l, 2 ≤ l ≤ n / 2.

For some fixed n, the POLYTOPE TRANSLATION problem for rational simplices Δ ⊂R^n is NP-hard.

If C is a nondegenerate hyperplane code, does T_C have a quadratic Gröbner basis? Does it have a quadratic generating set?

There exist constants K_1 and K_2 such that for every pair of integers m and n, where m ≥ K_1 and n ≥ K_2, we have χ_st'(C_m square P_n) = 6.

We don't think that our bound on the number of k-rich Möbius functions is tight. On the contrary, we conjecture that the right side can be replaced by O(n^4/k^3) in this case too.

This raises the question of how sparse a graph can be for the last Theorem to remain true.