Problems

Find other possible values of the parameter d and the corresponding d-antimagic labeling of type (1, 1, 1) for the hexagonal plane map H_n^m.

Are there families of graphs such that the independence equivalence class is unbounded and each independence polynomial is irreducible?

Working in the differential tower of groups A imathS with A abelian of order r, when k ≤n the critical group K(V(U^kD^k)_n)=K(Ind_A imathS_n-k^A imathS_n1) is given, as a list of elementary divisors,…

Every t-design of 2 t+k elements can be obtained from k points in t-good position using the methods developed here.

Let X,barψ be as in Lemma 6.2, and suppose that the roots of ∑_k=0^νp_kx^k are real. Then σ_ψ^2σ_X^2≥μ^4ν^-2 .

Let 3≤k≤ℓ and n≥2 ℓ . If G is an n-vertex k-chromatic ℓ -connected graph and t ≥ 3, then i_t(G)≤ i_t(G^*).

Is the Merino–Welsh conjecture true for binary matroids?

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.

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.