The covering threshold of a directed acyclic graph by directed acyclic subgraphs

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Let H be a directed acyclic graph (dag) that is not a rooted star. It is known that there are constants c = c(H) and C = C(H) such that the following holds for Dn, the complete directed graph on n vertices. There is a set of at most C log n directed acyclic subgraphs of Dn that covers every H-copy of Dn, while every set of at most c log n directed acyclic subgraphs of Dn does not cover all H-copies. Here this dichotomy is considerably strengthened. Let G(n, p) denote the probability space of all directed graphs with n vertices and with edge probability p. The fractional arboricity of H is (Formula presented) where the maximum is over all non-singleton subgraphs of (Formula presented) then H is totally balanced. Complete graphs, complete multipartite graphs, cycles, trees, and, in fact, almost all graphs, are totally balanced. It is proven that: • Let H be a dag with h vertices and m edges which is not a rooted star. For every a* > a(H) there exists c* = c*(a*, H) > 0 such a.a.s. G ~G(n, n−1/a*) has the property that every set X of at most c* log n directed acyclic subgraphs of G does not cover all H-copies of G. Moreover, there exists s(H) = m/2 + O(m4/5h1/5) such that the following stronger assertion holds for any such X: There is an H-copy in G that has no more than s(H) of its edges covered by each element of X. • If H is totally balanced then for every 0 < a* < a(H), a.a.s. G ~G(n, n−1/a*) has a single directed acyclic subgraph that covers all its H-copies. As for the first result, note that if h = o(m) then s(H) = (1 + om(1))m/2 is about half of the edges of H. In fact, for infinitely many H it holds that s(H) = m/2, optimally. As for the second result, the requirement that H is totally balanced cannot, generally, be relaxed.

Original languageEnglish
Article numberP4.45
JournalElectronic Journal of Combinatorics
Issue number4
StatePublished - 2022

Bibliographical note

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© The author. Released under the CC BY-ND license (International 4.0).

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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