Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

Abstract

Abstract. We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ = (1/2+α)n. For any constant α > 0, we give an optimal answer in the following sense: let reg even (n, δ) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. Then the number of edge-disjoint Hamilton cycles we find equals reg even (n, δ)/2. The value of reg even (n, δ) is known for infinitely many values of n and δ. We also extend our results to graphs G of minimum degree δ ≥ n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.

Edge-disjoint Hamilton cycles in random graphs

“…By (23) we can apply Lemma 38 with ε = 1/3, Q = V (G)\{x 0 } and S = S. But now N H −{x 0 } (S) = ∅, and so of the possible conclusions of Lemma 38 only (iv) can hold. This implies that |S| ≥ (n − 1)/6 > n/7.…”

Counting and packing Hamilton cycles in dense graphs and oriented graphs

“…For large graphs, Kühn and Osthus [24] further improved this to ⌊d/2⌋ edge-disjoint Hamilton cycles. Even more recently, after the first version of the present paper has been submitted, Csaba, Kühn, Lo, Osthus and Treglown [6] proved the exact version of the above conjecture for all large enough n.For the non-regular case, Kühn, Lapinskas and Osthus [22] proved that if δ(G) ≥ (1/2 + ε)n, then G contains at least reg even (n, δ(G))/2 edge-disjoint Hamilton cycles where reg even (n, δ) is the largest even integer r such that every graph G on n vertices with minimum degree δ(G) = δ must contain an r-regular spanning subgraph (an r-factor ). As for a concrete G, the maximal even degree r of an r-factor of G, which we denote by reg even (G), can be much larger than reg even (n, δ).…”

Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments

“…It would be interesting to obtain further conditions that relate the degree of the densest spanning regular subdigraph of a tournament T to the number of edge-disjoint Hamilton cycles in T . For undirected graphs, one such conjecture was made in [16]: it states that, for any graph G satisfying the conditions of Dirac's theorem, the number of edge-disjoint Hamilton cycles in G is exactly half the degree of a densest spanning even-regular subgraph of G. An approximate version of this conjecture was proved by Ferber, Krivelevich and Sudakov [12]; see, for example, [16,19] for some related results.…”