Problems

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

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

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.

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.

For any uncovered set S of patterns, fracR_S,nn converges in distribution to a Bernoulli random variable Ber(p) for some p ∈ [0, 1]. If S contains a pattern starting with 1, then p = 1, and if S contains a pattern starting with its largest…

The set of two-pop-stack sortable permutations of length 2n+1 with exactly n ascents has an equal number of permutations with last block of size one as permutations with last block size greater than one. That is, a(2n+1, n) = 2b(2n+1, n)…

What is the cop number of the flower snark J_n ?

For all k ≥ 0, the grid of size 3 × (2k+1) is an N-position for the game PUSH-CRAM. In addition, it is a bluff game.

For a simple example, consider the MINIMUM VERTEX COVER problem in fractionally treewidth-fragile graphs, or more generally in hereditary classes with sublinear separators. While the unweighted version can be dealt with by the local search…

Thus it remains to determine h_T(C_n) when n ≡ 1 or 2 (mod 3).

Take an n × n determinant such that its first column is the column of integers in the sequence of sequences S_i rectangular array and its first row is the k^th row, k=1,2,3,… . Then its determinant is given by (∏_j=1^n+1j^j-2)·( cn+k-1 n ).

Let r and q be integers such that 2 ≤ r ≤ 9 and r ≤ q. Then for all n ≥ r, the Turán graph T_r(n) has more q-colorings than any other graph with the same number of vertices and edges.

For every triangle-free graph G of minimum degree d, fracα(G)overlineα_G(1)≥ 2-o_d(1).

We have not been able to prove either way.

A balanced magic square of order 5 is completely balanced.

Is there a function f(x, y) such that for any graphs G_1 and G_2 we have χ_{td}(G_1 □ G_2) ≤ f(χ_{td}(G_1), χ_{td}(G_2))?

Is the reflexive dimension of the Minkowski sum P + P' bounded by refldim(P) + refldim(P') + c?

For any nonnegative integer k, does there exist a graph G that satisfies φ_(2,j)(G) − κ_(2,j)(G) + 1 = k?

What is the exact asymptotics of ex_e(G_1, n)?