Problems

For every k ≥ 1, there exists an integer f(k) such that every strong digraph with chromatic number greater than f(k) contains a subdigraph H with chromatic number at least k and such that H contains a Hamiltonian cycle.

Is being a length one TBU-poset (i.e. the covering graph has no 4-cycles) sufficient for cover preserving order embeddability in 2^n?

Let n ≥ 3 and let G be a 2-connected graph of order n with a nonnegative vertex weight function c. Then, μ_c(G)≤ n/4N/N-1 ifn is even, n/4N/N-1-N/4n(N-1) ifn is odd.

For every integer k ≥ 1, there exists an integer q=f(k) such that every k-ary tangram T can be factorized as T=X_1X_2··· X_q , so that the word U=X_a_1X_a_2··· X_a_q is a shuffle square, for some permutation σ=a_1a_2··· a_q .

For all nonempty sets S of patterns, the random variable T_S,n is asymptotically normal. In particular, fracT_S,n - E[T_S,n]Var(T_S,n) converges in distribution to a standard Gaussian as n → ∞. Furthermore, if S is uncovered, then…

σ(J_4)=0.28.

What is the generating function for partitions with profile segments of length less than 2, that is, into parts appearing not more than twice, with parts differing by at most 2, including starting with 1 or 2?

Suppose that ern(G) > 3 for a disconnected graph all of whose components are isomorphic to H. Then H is isomorphic to the star K_1,r where r is the number of edges.

A regular, k-intersecting hypergraph on n vertices has at most 2^{n-(2^{k+1}-k-2)} edges when k \ge 3.

Let G be a bipartite graph with sides A and B, and each edge colored red or blue. For a set X ⊆ A let N^RB(X) denote the set of vertices, which are joined to X by a red and by a blue edge as well. Suppose |N^RB(X)| ≥ |X| - 1 holds for…

The natural conjecture is that the Hasse index of any order h ≥ 1 is asymptotically Boolean, i.e. that i_h(D_n)=fracsc_h(D_n)|D_n|fracn^h2^h (for n →+∞ ) for every h ≥ 1.

For n<30 the maximal coefficient is not uniquely attained only for n=2,5,6,12,13,14and 15. Are those the only cases when this happens? If not, can we predict when?

Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is NP-complete, unless there exists h ∈ N such that A = 1, …, h.

Let n ≥k and t ≤ c = n/k. Let P ⊂ U_k^n be a partially t-intersecting partition system. Then, |P| ≤ (\begin{array}{c}n-tc-t\k-1\end{array}) U(n-c,k-1). Moreover, this bound is tight if and only if P is equal (up to permutations of [1, n])…

Let \lambda be a partition and \mu , and \eta be compositions such that |\lambda|=|\mu| and ll(\mu)\le |\eta| . Does there exist a quiver Q, dimensional vector \beta and GL(Q,\beta) -weight \sigma such that…

What is, for a given integer k ≥ 1 and any C (if k = 1, then C ≥ 1), the minimum m(C) such that any graph G with mad(G) ≤ 2k - m(C) satisfies χ_l(G^2) ≤ kΔ(G) + C.

It remains an open question as to whether ζ^*(n, S_7) grows faster than cubically.

In an intersecting r-partite hypergraph, what is the smallest size of a vertex cover that does not contain any edge or side?

Let b_t(n) denote the minimum number of edges induced by any set of n / 2 vertices in the Turán graph on n vertices for K_t .If each set of ⌊ n/2 ⌋ vertices in a graph G of order n spans more than b_t(n) edges, then G contains a K_t .

Conjecture 4.2. Let h≥1,(a_{1},...,a_{h})\in \mathbb{C}^{h} , and P(x)\in \mathbb{C}[x] . Set I_h,P,n(x)=P(x)∏_i=1^n(1+a_1x^F_i+a_2x^F_i+1+… +a_hx^F_i+h-1).Regarding h, P as fixed, let c_{n}(p) denote the coefficient of x^{p} in…