Manoussakis's pancyclicity conjecture for digraphs

Let DD be a strongly connected digraph on nn vertices. For vertices x,y,zV(D)x,y,z\in V(D), say that xx is nonadjacent to yy when neither arc between them is present. A digraph is pancyclic if it contains directed cycles of every length allowed by the conjecture.

Manoussakis's conjecture. If, for every triple x,y,zV(D)x,y,z\in V(D) with xx nonadjacent to yy, the inequalities

d(x)+d(y)+d+(x)+d(z)3n+1d(x)+d(y)+d^+(x)+d^-(z)\geq 3n+1

when xzA(D)xz\notin A(D), and

d(x)+d(y)+d+(z)+d(x)3n+1d(x)+d(y)+d^+(z)+d^-(x)\geq 3n+1

when zxA(D)zx\notin A(D), hold, then DD is pancyclic.

This is a proposed pancyclic analogue of Manoussakis-type Hamiltonicity degree conditions. The supplied source does not report a resolution.

Sources & referencesView supporting material

Primary source

Bo Ning, “Notes on a conjecture of Manoussakis concerning Hamilton cycles in digraphs”, arXiv:1404.5013 (2014).

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