Hamilton decompositions of regular expanders: A proof of Kelly’s conjecture for large tournaments

Abstract

A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n − 1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdős on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich. 1 arXiv:1202.6219v2 [math.CO] 10 May 2013It is not clear whether the lower bound on r in Theorem 1.1 is best possible. However, as discussed below, there are oriented graphs whose in-and outdegrees are all very close to 3n/8 but which do not contain even a single Hamilton cycle. Moreover, for r < (3n − 4)/8, it is not even known whether an r-regular oriented graph contains a single Hamilton cycle (this is related to a conjecture of Jackson, see the survey [36] for a more detailed discussion). Both these facts indicate that any improvement in the lower bound on r would be extremely difficult to obtain.Regular tournaments obviously exist only if n is odd, but we still obtain an interesting corollary in the even case. Suppose that G is a tournament on n vertices where n is even and which is as regular as possible, i.e. the in-and outdegrees differ by 1. Then Theorem 1.1 implies that G has a decomposition into edge-disjoint Hamilton paths. Indeed, add an extra vertex to G which sends an edge to all vertices of G whose indegree is below (n − 1)/2 and which receives an edge from all others. The resulting tournament G is regular, and a Hamilton decomposition of G clearly corresponds to a decomposition of G into Hamilton paths.The difficulty of Kelly's conjecture is illustrated by the fact that even the existence of two edge-disjoint Hamilton cycles in a regular tournament is not obvious. The first result in this direction was proved by Jackson [22], who showed that every regular tournament on at least 5 vertices contains a Hamilton cycle and a Hamilton path which are edge-disjoint. Zhang [48] then demonstrated the existence of two edge-disjoint Hamilton cycles. These results were improved by considering Hamilton cycles in oriented graphs of large in-and outdegree by Thomassen [46], Häggkvist [20], Häggkvist and Thomason [21] as well as Kelly, Kühn and Osthus [25]. Keevash, Kühn and Osthus [24] then showed that every sufficiently large oriented graph G on n vertices whose in-and outdegrees...

Edge-disjoint Hamilton cycles in random graphs

“…A very general result in this direction was obtained very recently by Kühn and Osthus [25], who showed that every regular 'robustly expanding' digraph of linear degree has a decomposition into edge-disjoint Hamilton cycles. The initial motivation for this result was that it implies that every (large) regular tournament has a Hamilton decomposition, which proved a long-standing conjecture of Kelly.…”

“…The initial motivation for this result was that it implies that every (large) regular tournament has a Hamilton decomposition, which proved a long-standing conjecture of Kelly. However, as observed in [26], the main result of [25] has a number of further applications. For instance, it can be used to show an analogue of Conjecture 1 for random tournaments, which confirms a conjecture of Erdős.…”

“…Our method is based on a new technique of iterative improvements which is likely to have further applications. In particular, this idea was an essential feature of the proof in [25].…”

Finding Hamilton cycles in robustly expanding digraphs

“…To see that the digraphs considered in Theorem 4 are robust outexpanders, we refer the reader to Lemma 11 of [22]. The fact that the graphs in Theorem 2(i) are robust outexpanders is proved in Lemma 12.1 of [21]. Lemma 6.2 of [15] shows that the oriented graphs considered in Theorem 2(ii) are outexpanders.…”

“…Theorem 5 is also used as a tool in a paper by Kühn and Osthus [21], which shows that if a digraph G is a robust outexpander whose minimum semi-degree is linear in n, then G has a Hamilton decomposition. More precisely, the fact that Theorem 5 is algorithmic is used in [21] to provide an algorithm which finds the above Hamilton decomposition in polynomial time.…”