@article{19503,
  abstract     = {A tantalizing open problem, posed independently by Stiebitz in 1995 and by Alon in 1996 and again in 2006, asks whether for every pair of integers  s,t≥1 there exists a finite number  F(s,t)
such that the vertex set of every digraph of minimum out-degree at least  F(s,t) can be partitioned into non-empty parts  A  and  B  such that the subdigraphs induced on  A
  and  B  have minimum out-degree at least  s  and  t , respectively.
In this short note, we prove that if  F(2,2)  exists, then all the numbers  F(s,t)  with  s,t≥1
  exist and satisfy  F(s,t)=Θ(s+t) . In consequence, the problem of Alon and Stiebitz reduces to the case  s=t=2 . Moreover, the numbers  F(s,t)  with  s,t≥2  either all exist and grow linearly, or all of them do not exist.},
  author       = {Christoph, Micha and Petrova, Kalina H and Steiner, Raphael},
  issn         = {1469-2163},
  journal      = {Combinatorics Probability and Computing},
  number       = {4},
  pages        = {559--564},
  publisher    = {Cambridge University Press},
  title        = {{A note on digraph splitting}},
  doi          = {10.1017/S0963548325000045},
  volume       = {34},
  year         = {2025},
}

@article{9591,
  abstract     = {We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.},
  author       = {Krivelevich, Michael and Kwan, Matthew Alan and Sudakov, Benny},
  issn         = {1469-2163},
  journal      = {Combinatorics, Probability and Computing},
  number       = {6},
  pages        = {909--927},
  publisher    = {Cambridge University Press},
  title        = {{Cycles and matchings in randomly perturbed digraphs and hypergraphs}},
  doi          = {10.1017/s0963548316000079},
  volume       = {25},
  year         = {2016},
}