All feedback ARC sets of a random turn tournament have [n/k] - k + 1 disjoint k-cliques (and this is tight)

Safwat Nassar, Raphael Yuster

Research output: Contribution to journalArticlepeer-review


What must one do in order to make acyclic a given oriented graph? Here we look at the structures that must be removed (or reversed) in order to make acyclic a given oriented graph. For a directed acyclic graph H and an oriented graph G, let fH(G) be the maximum number of pairwise disjoint copies of H that can be found in all feedback arc sets of G. In particular, to make G acyclic, one must at least remove (or reverse) fH(G) pairwise disjoint copies of H. Perhaps most intriguing is the case where H is a k-clique, in which case the parameter is denoted by fk(G). Determining fk(G) for arbitrary G seems challenging. Here we essentially answer the problem, precisely, for the family of k-partite tournaments. Let s(G) denote the size of the smallest vertex class of a k-partite tournament G. It is not difficult to show that fk(G) ≤ s(G) - k + 1 (assume that s(G) ≥ k - 1). Our main result is that for all sufficiently large s = s(G), there are k-partite tournaments for which fk(G) = s(G) - k + 1. In fact, much more can be said: A random k-partite tournament G satisfies fk(G) = s(G) - k + 1 almost surely (i.e., with probability tending to 1 as s(G) goes to infinity). In particular, as the title states, fk(G) =[ n/k] - k + 1 almost surely, where G is a random orientation of the Tur an graph T(n, k).

Original languageEnglish
Pages (from-to)1460-1477
Number of pages18
JournalSIAM Journal on Discrete Mathematics
Issue number2
StatePublished - 2021

Bibliographical note

Funding Information:
\ast Received by the editors July 30, 2020; accepted for publication (in revised form) March 21, 2021; published electronically June 22, 2021. Funding: This research was supported by the Israel Science Foundation, grant 1082/16. \dagger Department of Mathematics, University of Haifa, Haifa 31905, Israel (,

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics


  • Feedback arc set
  • K-clique
  • Random graph
  • Turn tournament

ASJC Scopus subject areas

  • Mathematics (all)


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