Finding Hamilton cycles in robustly expanding digraphs

Abstract

We provide an NC algorithm for finding Hamilton cycles in directed graphs with a certain robust expansion property. This property captures several known criteria for the existence of Hamilton cycles in terms of the degree sequence and thus we provide algorithmic proofs of (i) an 'oriented' analogue of Dirac's theorem and (ii) an approximate version (for directed graphs) of Chvátal's theorem. Moreover, our main result is used as a tool in a recent paper by Kühn and Osthus, which shows that regular directed graphs of linear degree satisfying the above robust expansion property have a Hamilton decomposition, which in turn has applications to TSP tour domination.

Hamiltonian degree sequences in digraphs

“…A (parallel) algorithmic version of Chvátal's theorem for undirected graphs was recently considered in [24] and for directed graphs in [8].…”

Hamilton cycles in dense vertex-transitive graphs

“…Yet in many situation there is an efficient algorithm for finding a Hamilton cycle in graphs satisfying certain conditions. See for example [7,26,9].…”

A Semiexact Degree Condition for Hamilton Cycles in Digraphs

“…Häggkvist [9] gave a construction showing that a minimum semidegree of 3n−4 8 is necessary and conjectured that it is also sufficient to guarantee a Hamilton cycle in any oriented graph of order n. This conjecture was recently proved in [11], following an asymptotic solution in [12]. In [5] we gave an NC algorithm for finding Hamilton cycles in digraphs with a certain robust expansion property which captures several previously known criteria for finding Hamilton cycles. These and other results are also discussed in the recent survey [17].…”

“…This point cannot be fully explained until we have given several definitions, but for the expert reader we make the following comment. Speaking very roughly, the general idea used in [12,11,5] is to apply Szemerédi's Regularity Lemma, cover most of the reduced digraph by directed cycles, and then use the expansion property guaranteed by the degree conditions on G to link these cycles up into a Hamilton cycle while absorbing any exceptional vertices. When the degrees are capped at n/2 two additional difficulties arise: (i) the expansion property is no longer sufficient to link up the cycles, and (ii) failure of a previously used technique for reducing the size of the exceptional set.…”

“…Our proof will use the machinery of Szemerédi's Regularity Lemma, which we describe in Section 4. (Unlike [12,11,5], we do not require the Blow-up lemma.) Section 5 contains an overview of the proof in a special case that illustrates the new methods that we introduce in this paper.…”