Cycles in dense digraphs

Abstract

Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let β(G) denote the size of the smallest subset XE(G) such that GX has no directed cycles, and let γ(G) be the number of unordered pairs {u, v} of vertices such that u, v are nonadjacent in G. It is easy to see that if γ(G) = 0 then β(G) = 0; what can we say about β(G) if γ(G) is bounded?

We prove that in general β(G) ≤ γ(G). We conjecture that in fact β(G) ≤ ½γ(G) (this would be best possible if true), and prove this conjecture in two special cases:

  • when V(G) is the union of two cliques

  • when the vertices of G can be arranged in a circle such that if distinct u, v, w are in clockwise order and uw is a (directed) edge, then so are both uv, vw.

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Correspondence to Maria Chudnovsky.

Additional information

This research was conducted while the author served is a Clay Mathematics Institute Research Fellow.

Supported by ONR grant N00014-04-1-0062, and NSF grant DMS03-54465.

This research was performed under an appointment to the Department of Homeland Security (DHS) Scholarship and Fellowship Program.

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Chudnovsky, M., Seymour, P. & Sullivan, B. Cycles in dense digraphs. Combinatorica 28, 1–18 (2008). https://doi.org/10.1007/s00493-008-2331-z

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Mathematics Subject Classification (2000)

  • 05C20
  • 05C35
  • 05C38