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CONNECTED TREEWIDTH can be solved by an O(n^f(k))-time algorithm.

K_{12t} minus a hamiltonian cycle K_{12t+3} minus a hamiltonian cycle K_{12t+8} plus a hamiltonian cycle can be decomposed into K_4's K_{12t+11} plus a hamiltonian cycle

A binary (or graphic) pregeometry of known cardinality and rank is reconstructible from its connected hyperplanes.

For any fixed k ≥0, the graphical Bell number sequence B(\overline{P_{n,k}}) is quasi geometric.

The minimum number of distinct abelian squares in a circular word of length n is: (a) (n-1)/2 if n is odd and this bound is attained only by a^n, a^n-1b and their complements and conjugates. (b) (n-2)/2 if n is even and this bound is…

If G is a graph with a dominating vertex, then aw(G, k) ≤ k + 1.

Let d ∈ [0, n] and E ⊆ [0, n]. For j ∈ [0, n], if j ∉ Z-cl_n,d(E), then there exists a polynomial P(X) = ℓ_1(X) … ℓ_k(X)σ(X) ∈ F_p[X], where deg ℓ_1 = … = deg ℓ_k = 1, and σ(X) is a symmetric polynomial, such that deg P ≤ d, P|_underlinei…

Next, can a design with λ > 1 be constructed using some but not all e-spaces, in which the lines of the design consist of all the lines of the underlying geometry? I conjecture that this is impossible.

(i) Is there an integer t such that every t-tough locally finite graph contains a Hamilton circle? (ii) Is there an integer t such that if deleting t k vertices from a locally finite graph G never leaves more than k infinite components…

For every k ≥ 3, we have r(k) = k.

Theorem 5.1, which produces infinitely many graphs with Δ≥4, leaves two open problems. The first is to determine whether the inequality in the theorem could be improved to a statement of equality.

It remains unknown to us whether or not there exists a divisional but not inductive poset among non-lattice, locally geometric posets.

Let G be a graph with maximum degree at most three. Suppose that G has an m-covering by 5-cycles, for some positive integer m. Then G is one of the seven graphs depicted in Figure 7.11.

Does the list of 5-capacities, presented in Section 4, include all members of GI_5 ?

Find an efficient algorithm to find L_{G+H}(\lambda,k)=max\left{L_{G}(\lambda,\ell)\cdots L_{H}(\lambda-\ell,k-\ell)\text{ for some }\chi(G)\leq \ell\leq k-1\right}.

Let G ≅∪_j=1^2KT_j be a disjoint union of an even number of path-like trees, all of them of the same order, and such that T_j≠P_2 for j=1,2, ..., 2 K. Is G a super edge-magic graph?

Let f(G) be the number of matchings of G. Is f(G)^1/|G|≥ f(H)^1/|H| (5.3) when H fractionally tiles G?

There are conjectured recurrences for T(m, n, k) (see A197654), but so far they are unproved.

For even value of m, it seems that rn(G(4mk+2m; 1,2m) = 2mk^2 + 2m^2k + 5mk + m^2 + m - k/2, if k is odd; 2mk^2 + 2m^2k + 7mk + m^2 + 2m - k - 1/2, if k is even.

Investigating Proposition 17, is there a more convenient expression for the upper bound based only on the Young diagram (see Figures 2 and 4) of the set of CRGs K(a, c) : H ↔_c K(a, c), ∀ H ∈ F(H)?