Approximate Hamilton decompositions of random graphs

Abstract

Abstract. We show that if pn log n the binomial random graph Gn,p has an approximate Hamilton decomposition. More precisely, we show that in this range Gn,p contains a set of edge-disjoint Hamilton cycles covering almost all of its edges. This is best possible in the sense that the condition that pn log n is necessary.

Edge-disjoint Hamilton cycles in random graphs

“…This will enable us to carry out rotations involving only the initial half of a long path in the proof of Lemma 43. The proof of Lemma 42 is similar to that of [20,Lemma 22].…”

“…But by (20) this implies that e H (S, T ) > 0. Hence H [Int F (V )] has only one component, which proves the claim.…”

“…Recall that in [20] we proved an approximate version of Conjecture 1, which finds a set of edge-disjoint Hamilton cycles covering all but an η proportion of the edges of G n,p . Roughly speaking, the proof of this approximate result proceeds as follows:…”

“…We now use a counting argument to show that P C (B) is small whenever C is large and B satisfies a very weak pseudorandomness condition. The proof builds on ideas from [12] and later developments in [24] and [20].…”

“…Noting that |V (G)\Q | ≤ 6|W | + 4 ≤ n/18000, we may apply Lemma 42 with ε = 1/15, H = H , U = Q and n = |P 1 |. Note that (14) is satisfied due to (20). Thus we obtain a set I ⊆ [log n] of size at least 14 log n/15 and sets J i ⊆ J i ∩ Q for each i ∈ I, such that |Int P 1 (J i )| ≥ 14|P 1 |/15 log n and the following holds: For every v ∈ J i , and for all but at most log n/15 indices j ∈ I,…”