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 $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.
Highlights
A directed graph D is a pair (V (D), E(D)) where E(D) ⊆ V (D)×V (D)
We prove a stronger statement which in some sense shows that the extremal condition is “stable,” i.e. graphs which do not satisfy the extremal condition do not require the tight minimum semi-degree condition
Note that there are less than 2|A|n ordered pairs that contain a vertex from A, so since λ α, we can greedily the electronic journal of combinatorics 22(4) (2015), #P4.34 choose vertex disjoint (xi, yi) ∈ fcon(di, ai+1) for each i ∈ [l − 1] such that xi, yi ∈/ A
Summary
A directed graph D is a pair (V (D), E(D)) where E(D) ⊆ V (D)×V (D). In this paper we will only consider loopless directed graphs, i.e. directed graphs with no edges of the type (v, v). In 1980, Grant made the weaker conjecture (replacing total degree by semi-degree) that if D is a directed graph on 2n vertices with δ0(D) n, D contains an ADHC [9]. Frye, Plantholt, and Tipnis conjectured that if D is a directed graph on 2n 8 vertices and δ0(D) n, D contains an anti-directed 2-factor [7]. Since it can be shown that Fn1 and Fn2 each contain an anti-directed 2-factor with two cycles we obtain the following corollary of Theorem 1.5, which implies their conjecture for sufficiently large n. Ln which corresponds to a spanning anti-directed ladder Ln in D
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