A Semiexact Degree Condition for Hamilton Cycles in Digraphs

Abstract

We show that for each β > 0, every digraph G of sufficiently large order n whose outdegree and indegree sequences d + 1 . . .is Hamiltonian. In fact, we can weaken these assumptions toand still deduce that G is Hamiltonian. This provides an approximate version of a conjecture of Nash-Williams from 1975 and improves a previous result of Kühn, Osthus and Treglown. 1 Proposition 7. Suppose G is a bipartite graph with vertex classes A and B and there is some number D such that for any S ⊆ A we have |N (S)| |S| − D. Then G contains a matching of size at least |A| − D.We will also need the following well-known fact.Proposition 8. Suppose that J is a digraph such that |N + (S)| |S| for every S ⊆ V (J). Then J has a 1-factor.Proof. The result follows immediately by applying Proposition 7 (with D = 0) to the following bipartite graph Γ: both vertex classes A, B of Γ are copies of the vertex set of the original digraph J and we connect a vertex a ∈ A to b ∈ B in Γ if there is a directed edge from a to b in J. A perfect matching in Γ corresponds to a 1-factor in J.We conclude by recording the Chernoff bounds for binomial and hypergeometric distributions (see e.g. [10, Corollary 2.3 and Theorem 2.10]). Recall that the binomial random variable with parameters (n, p) is the sum of n independent Bernoulli variables, each taking value 1 with probability p or 0 with probability 1 − p. The hypergeometric random variable X with parameters (n, m, k) is defined as follows. We let N be a set of size n, fix S ⊂ N of size |S| = m, pick a uniformly random T ⊂ N of size |T | = k, then define X = |T ∩ S|. Note that EX = km/n.

A survey on Hamilton cycles in directed graphs

“…. , n − 1 k times is both the out-and indegree sequence of G. In contrast to the undirected case there exist examples with a similar degree sequence to the above but whose structure is quite different (see [46] and [15]). This is one of the reasons which makes the directed case much harder than the undirected one.…”

Hamilton cycles in dense vertex-transitive graphs

“…We will only need this fact in the case that H is a matching. The proof of the above lemma given in [8] is probabilistic. (It proves that random subsets of sizes θm have the required property with high probability.)…”

Approximate Hamilton Decompositions of Robustly Expanding Regular Digraphs

“…The following lemma will use the special structure of the 1-factors to merge their cycles into a single Hamilton cycle. It is a special case of Lemma 6.5 in [13], which in turn is based on an idea in [6]. As noted in [13], the cycle guaranteed by the lemma can be found in polynomial time.…”