Manoussakis's pancyclicity conjecture for digraphs

About 12 years old · traced to

Let DD be a strongly connected digraph on nn vertices. For vertices x,y,z∈V(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,z∈V(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 xz∉A(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 zx∉A(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.

References

Primary source

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

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.