Problems

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)?

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.

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

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 ?

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|?