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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 .

Let m > n, and let b = v_i_i=1^n be a basis for Z^n. For a finite index subgroup Σ < Z^n consider the subset S_Z^n/Σ(b, m) = s ∈ S_Z^n/Σ(m) : s ⊃ b mod Σ (that is, we restrict attention to the generating sets which contain the reduction of…

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?

Does this set contain interior points within the manifold A ∈ R_+^m × n | rk A = k of nonnegative rank-k matrices?

If we remove the equal coefficients condition, does Theorem 4.1 hold with n = d + 1?

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.

(i) For every triangulation K of a Witt space with vanishing middle intersection homology GIN(K) ⊂ GIN(C(d,n)). (2.1) (ii) The strong upper bound conjecture holds for arbitrary polyhedral complexes (and even for all regular cell complexes…

There exists t_0 ∈ N such that m_G_t(-∞, -2) is constant for all t ≥ t_0.

If vecG is a best-balanced orientation of G := (V + s, E) and varrho_vecG(s) = δ_vecG(s) then there exist rs, st ∈ A(vecG) so that vecG_rt is a best-balanced orientation of G_rt.

For any positive integers k and n satisfying k < n, and any alternating function f: [k] × [k] → Z_n, there exists a permutation π ∈ S_k such that d_π(i, j) ≠ f(i, j) pmodn, for all distinct i, j ∈ [k].

We don't know if the theorem 2 is true whenever p=ω , m=ω_1,n=ω_2 and n^p=ω_3=2^p : we do not suppose g.c.h. .

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

Let f be a finite field. Suppose barX, barY, barα, and barZ_f are random variables with values in f; barα is distributed with respect to the probability counting measure on the set f^× of non-zero elements of f, and barZ_f is distributed…

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.