Semi-degree threshold for anti-directed Hamiltonian cycles

Abstract

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2, then D contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph D to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even n, if D is a directed graph on n vertices with minimum out-degree and in-degree at least n 2 + 1, then D contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that n 2 is sufficient unless D is one of two counterexamples. This result is sharp.

Arbitrary Orientations of Hamilton Cycles in Digraphs

“…This improved a result of Grant [5] for antidirected Hamilton cycles. The exact threshold in the antidirected case was obtained by DeBiasio and Molla [2], here the threshold is δ 0 (G) ≥ n/2 + 1, i.e., larger than in Ghouila-Houri's theorem. In Figure 1, we give two digraphs G on 2m vertices which satisfy δ 0 (G) = m and have no antidirected Hamilton cycle, showing that this bound is best possible.…”

Packing and counting arbitrary Hamilton cycles in random digraphs

“…Later on, Häggkvist and Thomason [15] showed an approximate analog of the result of Ghouila-Houri [13] while proving that 0 (G) ≥ n∕2 + n 5∕6 is sufficient to guarantee every orientation of a Hamilton cycle appears in G. Very recently, this problem has been settled completely by DeBiasio and coworkers [4]. They showed that 0 (G) ≥ n∕2 is enough for all cases other than an anti-directed Hamilton cycle, where for the latter, Debiaso and Molla showed in [5] that 0 (G) ≥ n∕2 + 1 is enough (an anti-directed Hamilton cycle is a cycle with no two consecutive edges having the same orientation).…”

“…Very recently, this problem has been settled completely by DeBiasio and coworkers. They showed that δ 0 ( G ) ≥ n /2 is enough for all cases other than an anti‐directed Hamilton cycle, where for the latter, Debiaso and Molla showed in that δ 0 ( G ) ≥ n /2 + 1 is enough (an anti‐directed Hamilton cycle is a cycle with no two consecutive edges having the same orientation).…”

On Degree Sequences Forcing The Square of a Hamilton Cycle

“…This will allow us to apply Lemma 6.3 to G[R] rather that G itself. The idea of connecting paths through a reservoir has been used, for example, in [13,17,33,37].…”