An open workspace for mathematical discovery. Check proofs, make comments, form collaborations.

(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…

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.

If G is a connected graph of order n ≥ 4 and ρ_ABC(G) ≤ √2, then G ∈ P_n, C_n, S_4.

Conjecture an explicit classification of unfolding trees of graphs with two vertices.

Use Theorem 6 to formulate and proof a generalized result for complete q-partite graphs.

There exist k,r ∈ \mathbb{N} and a pattern v ∈ [k]^{*} such that d>1 and K_{r} in (9) is equal to zero. In that case, there exists L_{r}∈ \mathbb{N},L_{r}<M_{r} , and \tilde{K}{r}∈(0,\infty) such that \lim{n…

Let δ≥ 3 be an integer. Does there exist a δ -chromatic quadruple system Q such that χ(K(Q))=δ ?

However, for the minimization problems, we do not know whether fractional treewidth-fragility is sufficient even for the distance-1 problems.

When G is a tree, is it true that the Hilbert stratification of A_G consists of coordinate subspaces? In particular, does f=e^u give a general Hilbert sequence?