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Is it possible to embed, a given unicyclic graph in a graceful unicyclic graph?

It would be of interest to determine whether the growth of \sup_{i\le n}NMC(i) is exponential as n \to\infty or if the subsequence of strictly increasing terms is exponential.

For any positive integer t, is there any bipartite graph G such that dis_ℓ[G] - dis[G] ≥ t?

Show that 2 ·1_k-1 and 2 ·0_k-1 are the only (k-1)-rowed critical sub structures of K_k .

Let V be a 2-dimensional subspace in F_p^4, such that 1 ∈ V. Then V is a clique in PP(p^4, p + 1, I) for some I if and only if V = F_p ⊕ aF_p, where a = g^(p+1)k and k is an odd integer.

Let b: V → Z_0^+ be a symmetric crossing submodular function with b(∅) = 0 and b(X) ≡ |X ∩ T_b| mod 2. Then there exists a pairing M on T_b that satisfies (17).

Does our improved estimate for the edge isoperimetric inequality in the remark following Theorem 3.1 hold for general B, i.e. do we always have ∂_e,G_B(A)≥ d μ(Z(B))^1/d|A|^1-1/d ?

Let \lambda be a partition, \mu and \eta be compositions such that |\lambda|=|\mu| and ll(\mu)\le |\eta| . Then the coefficient c(\lambda,\mu | \eta) is a homogeneous piecewise linear function of \lambda and \mu. In particular, c(N\lambda,…

For any triple of positive integers a = (a_1, a_2, a_3) the sequence of numbers the sequence g_i(a)_i=0^∞ is monotonically increasing with i, for i ≤ 14.

More generally, call a set of subgraphs of G a packing if the subgraphs are disjoint. Let f(G) be the number of packings of G by copies of a fixed graph K (so when K is an edge, this is the number of matchings). Does (5.3) hold when H…

Identify and characterise the product graphs whose curling numbers are the product of the curling numbers of their factors graphs.

There is no graph with uniform rank spread equal to two.

Every k-ary tangram T satisfies cut(T, R_k) ≤ d_k, for some finite constant d_k depending only on k.

If a neural code is labeled such that the i^th neuron is added as a piercing at the i^th step, then its toric ideal has a quadratic Gröbner basis with respect to the term order prec.

The set reconstruction problem for multisets over commutative groupoids is an open problem, and it may be worth investigating.

Assume that p nmid (m-1) and 2 ≤ m ≤ q+1/3. Then the cliques from Proposition 4.6 and Proposition 4.7 are maximal.

Does there exist a graph with a constant neighbourhood of two independent vertices containing the maximal independent vertex sets V1 of cardinality k1 and V2 of cardinality k2 such that k1 ≠ k2 ?

From the above theorem, it seemed to suggest the following conjecture: if G_1 is m-extendable and G_2 is n-extendable (m, n ≥ 0), then their lexicographic product G_2 ∘ G_1 is (n+1)|V(G_1)|-factor-critical.

Specifically, we conjecture that similar processes will work for cycle pendant stars up to stars of size 15.

The generating function of the colored excedance on SD_n^D is log-concave for all n ≥ 1. The generating function of the absolute excedance on SD_n^D is log-concave for all n ≥ 10 and unimodal for all n ≥ 1.