Optimal covers with Hamilton cycles in random graphs

Abstract

Abstract. A packing of a graph G with Hamilton cycles is a set of edgedisjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in Gn,p a.a.s. has size δ(Gn,p)/2 . Glebov, Krivelevich and Szabó recently initiated research on the 'dual' problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for log 117 n n ≤ p ≤ 1 − n −1/8 , a.a.s. the edges of Gn,p can be covered by ∆(Gn,p)/2 Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szabó, which holds for p ≥ n −1+ε . Our proof is based on a result of Knox, Kühn and Osthus on packing Hamilton cycles in pseudorandom graphs.

Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

“…Indeed, the results of several authors (mainly Krivelevich and Samotij [10] as well as Knox, Kühn and Osthus [9]) can be combined to show that for all 0 p 1, the binomial random graph G n,p contains δ(G n,p )/2 edge-disjoint Hamilton cycles with high probability. Some further related results can be found in [6,11,12].…”

Edge-disjoint Hamilton cycles in random graphs

“…The main result in shows that this bound is approximately true if . Subsequently, Hefetz, Kühn, Lapinskas and Osthus achieved an exact version of this result for a similar range of p as in Theorem . Theorem Let .…”

“…For (a), (b) and (c) we will prove slightly stronger bounds, as these will be used in [17]. We will need the following large deviation bounds on the binomial distribution, proved in [18] (as Theorem 2.1, Corollary 2.3 and Corollary 2.4 respectively):…”

On the Number of Hamilton Cycles in Sparse Random Graphs

“…Recent results include packing and covering problems (see e.g. [13], [21], [22], [26], [15], and [18]), local resilience (see e.g. [34], [14], [4] and [28]) and Maker-Breaker games ( [33], [16], [3], and [11]).…”