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Is it possible to prove an analogue of Theorem 3.1 for non-uniform hypertrees?

Find a function a(k) such that for every k we have the tight bound γ_krt(G) ≤ a(k) · γ_rk(G).

Let G be a 2k-edge-connected graph with k ≥ 1 and let L(v) ⊆ k, …, d_G(v) such that |L(v)| ≥ ⌈ d_G(v)/2 ⌉ - ⌈ k/2 ⌉ + 2 for every v ∈ V(G). Is it true that G includes a k-edge-connected L-factor?

Consider maximal planar graphs, 3-connected planar graphs or planar 3-trees. For all such graphs G, is there a constant α<1 and a constant k such that Z(G)≤ α n+k ?

For d ≥ 3, is every minimum edge cut of a flag d-polytope or a balanced d-polytope trivial?

(1) χ(G(2,7,5))=4?

Is every wqo class of graphs in fact bqo?

Let G* be a maximum bipartite minor of a graph G as defined in Thm. 3.29. Is there a generalized Laplacian matrix M(G) such that an eigenfunction of M(G) has |V(G*)| weak nodal domains?

For some ε > 0 the following holds. Let A_i = p_i, q_i, i = 1, 2, 3, 4, be four sets of two points each in R^3, such that |p_i - q_i|_2 < ε. Then the set of midpoints between different A_i, bigcup_i,j=1,2,3,4, atop i ≠ j 1/2(A_i + A_j), is…

If G ∈ G_k with k ≥3 is a graph with maximum degree Δ ≥frac(2k-1)^2k-1 ,then G is equitably m-colorable for every m≥Δ .

When t=1 and k>1, ⟨ P_n^k(x;t),∑_μ vdash k*ns_μ(x)⟩ equals one of the numbers 0,1,2,…,n-1.

For any k \ge 3 and for any p, there exists a finite number of quasi-strongly regular graphs of grade p.

We conjecture that the infinite family of appended graphs has unique betweenness centrality.

The expression α^ℓ(μ)-1H_μP_μ^#(x_1,...,x_n) has nonnegative coeffi cient in the basis (α^c(x_1-x_2)_b_1… (x_n-1-x_n)_b_n-1(x_n)_b_n)_c,b_1,...,b_n≥ 0,where (x)_b is as usual the b^th falling power of x, that is x(x-1)… (x-b+1) .

Let G_k = ([k], E_k) be the complete k-hygraph on [k]. Given a k-hygraph G([n], E), (easily connected?), consider any function f: [n] → [k] and extend it naturally to f: E → E_k, and define G_f := ([k], f(E)). Define further φ: Q^n → Q^k…

For any graph G on n ≥ 4 vertices, ▷ ∂{2}^{L}(G)≥n with equality if and only if G is the complete graph K{n} or K_{n} minus an edge; ▷ if n ≠7, then ∂{2}^{L}(G)≤∂{2}^{L}(P_{n}) with equality if and only if G is the path P_{n} ; ▷ if G is a…

Determination of ξ_G(λ_χ) for general chromatic characteristic polynomials of all 2-regular bipartite graphs is still in progress.

CONJECTURE M(D) : For all A ∈∂ Ω(R_n),f_A(λ)=per(λ A+(1-λ)D_n) is nondecreasing in the interval 0 leqqλ leqq 1 .

For each positive integer t, is there a bipartite graph G such that V(G) = O(t^c) and dis[G] > t, where c is a constant number.

Let k_1, …, k_n-1 be a sequence of nonnegative integers and let M = x_12^k_1x_23^k_2 … x_n-1,n^k_n-1. Then the reduced form of M evaluated at x = (1, …, 1) and β-1 in tildeACYB_n(β) is a polynomial in β with nonnegative coefficients.