We show that with high probability the random graph G n,1/2 has an induced subgraph of linear size, all of whose degrees are congruent to r (mod q) for any fixed r and q ≥ 2. More generally, the same is true for any fixed distribution of degrees modulo q. Finally, we show that with high probability we can partition the vertices of G n,1/2 into q + 1 parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to r (mod q). Our results resolve affirmatively a conjecture of Scott, who addressed the case q = 2.
We show that with high probability the random graph Gn,1false/2$$ {G}_{n,1/2} $$ has an induced subgraph of linear size, all of whose degrees are congruent to r0.3emfalse(mod0.3emqfalse)$$ r\kern0.3em \left(\operatorname{mod}\kern0.3em q\right) $$ for any fixed r$$ r $$ and q≥2$$ q\ge 2 $$. More generally, the same is true for any fixed distribution of degrees modulo q$$ q $$. Finally, we show that with high probability we can partition the vertices of Gn,1false/2$$ {G}_{n,1/2} $$ into q+1$$ q+1 $$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to r0.3emfalse(mod0.3emqfalse)$$ r\kern0.3em \left(\operatorname{mod}\kern0.3em q\right) $$. Our results resolve affirmatively a conjecture of Scott, who addressed the case q=2$$ q=2 $$.
For $$0\le \ell <k$$ 0 ≤ ℓ < k , a Hamilton $$\ell $$ ℓ -cycle in a k-uniform hypergraph H is a cyclic ordering of the vertices of H in which the edges are segments of length k and every two consecutive edges overlap in exactly $$\ell $$ ℓ vertices. We show that for all $$0\le \ell <k-1$$ 0 ≤ ℓ < k - 1 , every k-graph with minimum co-degree $$\delta n$$ δ n with $$\delta >1/2$$ δ > 1 / 2 has (asymptotically and up to a subexponential factor) at least as many Hamilton $$\ell $$ ℓ -cycles as a typical random k-graph with edge-probability $$\delta $$ δ . This significantly improves a recent result of Glock, Gould, Joos, Kühn and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values $$0\le \ell <k-1$$ 0 ≤ ℓ < k - 1 .
For 0 < k, a Hamiltonian -cycle in a k-uniform hypergraph H is a cyclic ordering of the vertices of H in which the edges are segments of length k and every two consecutive edges overlap in exactly vertices. We show that for all 0 < k − 1, every k-graph with minimum co-degree δn with δ > 1/2 has (asymptotically and up to a subexponential factor) at least as many Hamiltonian -cycles as in a typical random k-graph with edgeprobability δ. This significantly improves a recent result of Glock, Gould, Joos, Kühn, and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values 0 < k − 1.
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