Minimum total-degree threshold for forcing the k-th power of a Hamilton cycle in a digraph

Establish that for every fixed integer k ≥ 1 and all sufficiently large integers n, writing n = (k+3)q + r with q ∈ Z and r ∈ {0,1,…,k+2}, any n-vertex digraph G with minimum total degree δ(G) at least 2(1 − 1/(k+3))n − 3 if r = k+2, at least 2(1 − 1/(k+3))n − 2 if r ∈ {k, k+1}, and at least 2(1 − 1/(k+3))n − 1 otherwise, must contain the k-th power of a Hamilton cycle.

Background

The paper studies degree thresholds that force powers of Hamilton cycles in directed settings. In undirected graphs, the Pósa–Seymour conjecture (now a theorem for large graphs) determines the exact minimum degree threshold for the k-th power of a Hamilton cycle. For digraphs, analogous results are much less developed. The authors propose a precise minimum total-degree threshold that depends on n modulo (k+3) for guaranteeing the k-th power of a Hamilton cycle in an n-vertex digraph.

They provide an extremal construction showing the conjectured bounds are best possible and prove an asymptotic version for the square (k=2) of a Hamilton cycle. Extending this to all k remains open and appears to hinge on developing stronger connecting lemmas in digraphs.