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For an integer k ≥ 2 and a sufficiently large n. Let G be an n vertex C_3ℓ+1-free graph for every integer ℓ ≥ k. Then for every r, 3k ≥ r ≥ 2, the number of cliques of size r in G is at most n-1/3k-1 binom3kr. Equality holds only for…

If we assume this conjecture, then we show that 1/k!∑_c ∈ LQ(m,k)ε(c)≥ 0 . This is still an open problem.

We conjecture that any fixed value occurs finitely many times.

To replace this conjecture, a new conjecture is formulated that we coin as the Brualdi-Li tally conjecture.

We conjecture that any such A can be covered by O(1) homothetic copies of C(B) with total volume O(|A|).

Let G be an (n, λ)-connected graph and let S be a given subset of V(G) such that |S| = n. Then G has λ edge-disjoint cycles C_1, ..., C_λ such that S ⊆ V(C_i) for all i, 1 ≤ i ≤ λ.

Let G' be a graph obtained from a (Δ,1) -bidegreed graph G by inserting a bouquet in an edge of a bouquet internal path. Then (i) if ρ(G)<1+√Δ-1, then ρ(G^′)>ρ(G); (ii) if ρ(G)>1+√Δ-1, then ρ(G^′)<ρ(G); (iii) if ρ(G)=1+√Δ-1, then…

It would thus be natural to conjecture that T(H) = Θ(n!/2^e(H)) for, say, all (ε, k)-consistent orientations with n vertices.

K(n, m) \le \lceil (m-1)d^2/n + 1/m \rceil, where d = \lfloor n/m \rfloor.

The condition in Theorem 4 is necessary as well as sufficient.

Does Theorem 1 hold with no restriction on K? If not, what is the least information needed on K?

The Q-Kostka polynomials L_λ μ(q) have non-negative coefficients.

Can we find the minors for this property?

Open problem: • d∈0,2 for n>7

We leave as Conjecture 5.38 that this also holds for n even.

What is the computational complexity of COMPLETE WIDTH on 2K_2 -free graphs?

it is not known whether \alpha_{\lambda}(k;p) has unimodal coefficients for all \lambda and k.

Clearly a(v) \le \bar{a}(v) and we conjecture that a(v) = \bar{a}(v) based on empirical observations.

The element tildew_b is maximal in the weak order on tildeW/W among all dominant elements tildew ∈ tildeW/W : tildew^-1(0) ∈ S(b).

Pascal-type behavior except for the entry 14.