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Given a p × q integer matrix M with p ≥ 2, if none of the differences between two rows of M is parallel to 1^{T} , then m(M,n)=(2+o(1))n/log_{p}n.

Suppose now that b ≤ a < s, and m ≫ n^1+s-1. Then satex(n, K_1,s : m, K_a,b) = (1 + o(1)) minN(K_a,b, K_q^*), N(K_a,b, overlineK_r^*), where q = mint ∈ Z : N(K_1,s, K_t) ≥ m and r = mint ∈ Z : N(K_1,s, overlineK_t) > m.

(Brill-Noether Existence for ℝ-Divisors on Graphs) Let ρ(g,r,d)=g-(r+1)(g-d+r). Fix two real numbers r ≥0, 2 g-2 ≥d. If ρ(g,r,d)≥0 then there exists an ℝ-divisor of degree at most d and rank equal to r on G.

The family of graphs G_4,0,r, where r ≠ c^3 + 4c^2 + 1 + q and c ∈ N, excluding the graphs G_4,0,4, G_4,0,9, G_4,0,10 and G_4,0,17, is a family of Galois equivalent graphs with each P(G_4,0,r, λ) having Galois group S_4.

Conjecture 3.7. Let k be a positive integer, G=(V,E) be a graph, and r:V \to Z_{+} such that r(V) ≥k+1. Then G has a k-connected r-detachment if and only if (a) G-Y is (k-r(Y))-edge connected for all Y \subseteq V with r(Y) ≤k-2, (b)…

Characterize the graphs G such that every irreducible dominating set in G is either a minimal dominating set or a minimal total dominating set.

It is an important problem to determine if equality holds, since it would give an intrinsic description of the amplituhedron which does not mention Gr_k,n^≥ 0.

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.