Minimum total-degree threshold for k-th powers of Hamilton cycles in digraphs

Establish that for every fixed integer k ≥ 1 there exists n0 such that, for all integers n ≥ n0 with n = (k+3)q + r where q ∈ Z and 0 ≤ r ≤ k+2, every n-vertex digraph G with minimum total degree δ(G) at least 2(1 − 1/(k+3))n − 3 when r = k+2, at least 2(1 − 1/(k+3))n − 2 when r = k or r = k+1, and at least 2(1 − 1/(k+3))n − 1 otherwise, contains the k-th power of a Hamilton cycle.

Background

This conjecture proposes the precise minimum total-degree threshold in general digraphs that forces the existence of the k-th power of a Hamilton cycle. It is the directed analogue of the Pósa–Seymour threshold in undirected graphs, adjusted for the directed setting via an extremal construction that depends on n modulo k+3.

The paper proves the asymptotic version for k=2 (squares of Hamilton cycles) but leaves the general k case open. The authors also provide an extremal construction showing the proposed piecewise bounds on δ(G) would be tight if true.