Problems

An interesting open problem is to find all the values that can be attained from paths.

We conjecture that a 3-edge-connected, nonplanar graph with representativity at least 5 has exponentially many peripheral cycles.

Open problem: • d ∈ 3, 4 for n ≥ 1

The triple (L_([2] × [2]) × [2](R^a+3), Pro, ξ(f, (x_1, x_2), a) + ξ(f, (3-x_1, 3-x_2), a)) is 4-mesic.

It seems reasonable to conjecture that for degree sequences of any order n ≥ 1 the modal multiplicity will be 1, while the median and mean multiplicities will both increase with n, the latter much more rapidly than the former.

Here is another variant that is open. In this case we begin with a -1, +1 valued matrix with discrepancy n^3/2. Say a Hadamard matrix. Balancer picks certain +1's. Unbalancer picks certain -1's. Over the course of the game, can Balancer…

Is it true for every t that ¿ lim_n →∞F(n;t)/n=1/2?

We formulate an analogue of Conjecture 1.2 for term orders with x_1 > x_2 > … > x_n (Conjecture 11.15).

For an integer k ≥ 2 and a sufficiently large n. Let G be an n vertex C_3ℓ+1-free graph for every integer ℓ ≥ k. Then for every r, 3k ≥ r ≥ 2, the number of cliques of size r in G is at most n-1/3k-1 binom3kr. Equality holds only for…

If we assume this conjecture, then we show that 1/k!∑_c ∈ LQ(m,k)ε(c)≥ 0 . This is still an open problem.

We conjecture that any fixed value occurs finitely many times.

To replace this conjecture, a new conjecture is formulated that we coin as the Brualdi-Li tally conjecture.

We conjecture that any such A can be covered by O(1) homothetic copies of C(B) with total volume O(|A|).

Let G be an (n, λ)-connected graph and let S be a given subset of V(G) such that |S| = n. Then G has λ edge-disjoint cycles C_1, ..., C_λ such that S ⊆ V(C_i) for all i, 1 ≤ i ≤ λ.

Let G' be a graph obtained from a (Δ,1) -bidegreed graph G by inserting a bouquet in an edge of a bouquet internal path. Then (i) if ρ(G)<1+√Δ-1, then ρ(G^′)>ρ(G); (ii) if ρ(G)>1+√Δ-1, then ρ(G^′)<ρ(G); (iii) if ρ(G)=1+√Δ-1, then…

It would thus be natural to conjecture that T(H) = Θ(n!/2^e(H)) for, say, all (ε, k)-consistent orientations with n vertices.

K(n, m) \le \lceil (m-1)d^2/n + 1/m \rceil, where d = \lfloor n/m \rfloor.

The condition in Theorem 4 is necessary as well as sufficient.

Does Theorem 1 hold with no restriction on K? If not, what is the least information needed on K?

The Q-Kostka polynomials L_λ μ(q) have non-negative coefficients.