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

Considering the graphs H_3(d) leads to the conjecture of e(G^3)≥ 2e(G),for G regular, connected, and diam(G) ≥ 3.

For even m and n > m ≥ 4, Δ(T_n) ≤ n - m + 2 ⇒ R(T_n, W_m) = 2n - 1.

For positive integers k, d and n with k ≥3, find the largest value f_{k,d}(n) such that every connected graph G of maximum degree at most d and of order n contains a k-tree T with |T|≥f_{k,d}(n).

In PG(3, q) and PG(4, q), the upper bounds (1.5), (1.6) hold for all q.

(Covering Radius Conjecture) Let λ ∈ [1/g, g] (recall that g ≥ 1). The covering radius of N_G with respect to the polytope P_{1,λ} is at least √(g/λ)/n where n is the number of vertices of G.

A set S of patterns is uncovered if and only if it satisfies f_n = O(t_n+1).

If r is odd, a basis of MR^sharp will be parametrized by colored compositions such that parts of color 0 are not ≡ 0 bmod r and parts of color 1 are arbitrary. The Hilbert series is then H_r(t)=frac1-t^r1-2(t+t^2+… +t^r). If r is even,…

If G is a critical graph (not banned) and ρ^⊥(G)=d , then G-M(G) (with M a match maximal) has ρ^⊥(G-M(G))≤ d-1

Perhaps an equally daring conjecture would be that L(G) = {d : d divides |G|}, in which case we would have f(G) = f^{*}(|G|).

Let Δ be a Nim-regular complex, F a nonempty face, D_i a minimal cover of F by circuits and D = uplus_i D_i. Is it necessarily true that D - F is not a DUOC?

How can we read the combinatorics of the point configuration S ⊂ TT^X from the split system of C_S ?

Can one prove better upper or lower bounds for ℓ_n ?

Conjecture 1. Given an undirected connected graph G. We consider v an extremum of the Fiedler vector of the graph G.\tilde{G} is the graph obtained from G and v as in Proposition 1. Then for all x>0 the Fiedler vector \Phi(x,\cdot) of…

Is it true that for every nonnegative integer k, there exists a connected hypergraph H satisfying φ_ST(H) − κ_ST(H) + 1 = k?

Let G be a 2-connected bipartite graph with sides A and B satisfying |N^2(X)| ≥ |X| for every X ⊆ A of size at least 3. For every X ⊆ A, |X| ≥ 3, there is a cycle C_X in G such that V(C_X) ∩ A = X.

Let G be a fuzzy graph. Then, (1) s(G) ≤ 2Δ(G) + 1, (2) for every integer k ≥ 2, there exists a fuzzy graph G_k such that k-1 ≤ Δ(G_k) and s(G_k) = Δ(G_k) + k.

Let M(n) denote the absolute value of the Möbius function μ[1,W_n]=μ[1,M_n] . Then for n>50 we have M(2n)=n^2⇔ n+1 isprimeandn ≡ 0(bmod6) M(2n)=n^2-1 ⇔ n+1 isprimeandn ≡ 4(bmod6) M(2n+1)=n^2-n ⇔ n+1 isprimeandn ≡ 0(bmod6) M(2n+1)=n^2-n-1 ⇔…

Let G be a connected graph of order n. If 1/2<α<1 , then λ_n(A_α(G))≥ λ_n(A_α(K_1,n-1)),the equality holds if and only if G ≅ K_1,n-1 .

First of all, is it possible to sharpen Theorem 1 to the assertion that ‘colex is best’: if A ⊂ [n]^(3), and C is the set of the first |A| elements of [n]^(3) in the colex order, then must we have |VA| ≥ |VC|?

Suppose that G ∈ G^r . Is λ_min^(p)(G) continuously differentiable for p>r ? Is λ_min^(p)(G) continuously differentiable for p \ne k, k=2, ..., r ?