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 X ⊆ E(G) such that G∖X 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:
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when V(G) is the union of two cliques
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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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References
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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.