Problems

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.

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?

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

Pascal-type behavior except for the entry 14.

For every k ≥ 1, there exists an integer f(k) such that every strong digraph with chromatic number greater than f(k) contains a subdigraph H with chromatic number at least k and such that H contains a Hamiltonian cycle.

Is being a length one TBU-poset (i.e. the covering graph has no 4-cycles) sufficient for cover preserving order embeddability in 2^n?

Let n ≥ 3 and let G be a 2-connected graph of order n with a nonnegative vertex weight function c. Then, μ_c(G)≤ n/4N/N-1 ifn is even, n/4N/N-1-N/4n(N-1) ifn is odd.

For every integer k ≥ 1, there exists an integer q=f(k) such that every k-ary tangram T can be factorized as T=X_1X_2··· X_q , so that the word U=X_a_1X_a_2··· X_a_q is a shuffle square, for some permutation σ=a_1a_2··· a_q .

For all nonempty sets S of patterns, the random variable T_S,n is asymptotically normal. In particular, fracT_S,n - E[T_S,n]Var(T_S,n) converges in distribution to a standard Gaussian as n → ∞. Furthermore, if S is uncovered, then…

σ(J_4)=0.28.

What is the generating function for partitions with profile segments of length less than 2, that is, into parts appearing not more than twice, with parts differing by at most 2, including starting with 1 or 2?

Does this set contain interior points within the manifold A ∈ R_+^m × n | rk A = k of nonnegative rank-k matrices?