Hamilton cycles in random graphs with minimum degree at least 3: An improved analysis

Abstract

In this paper we consider the existence of Hamilton cycles in the random graph G = G ≥3 n,m. This random graph is chosen uniformly from  ≥3 n,m , the set of graphs with vertex set [n], m edges and minimum degree at least 3. Our ultimate goal is to prove that if m = cn and c > 3∕2 is constant then G is Hamiltonian w.h.p. In Frieze (2014), the second author showed that c ≥ 10 is sufficient for this and in this paper we reduce the lower bound to c > 2.662 …. This new lower bound is the same lower bound found in Frieze and Pittel (2013) for the expansion of so-called Pósa sets.

Fast algorithms for solving the Hamilton Cycle problem with high probability

“…n,m be a random graph that is chosen uniformly from all the graphs on [n] with m edges and minimum degree 2. What follows is taken (with minor modifications) from [4], [5], [12].…”

“…The following Lemma allows us to treat G i as a random graph of minimum degree 2. For its proof see [4], [5], [12].…”

“…This fact is used in Lemma C.4 which establishes concentration of the number of vertices of degree k in J 2 . For the proofs of Lemmas C.3 and C.4 see [5], [4], [12]. We let…”

“…By slightly modifying the arguments given in [4], [5] and [12], one can prove Lemma 3.11. For sake of completeness we display the key elements of the corresponding arguments at Appendix C. Lemma 3.11.…”

Packing Hamilton Cycles in Cores of Random Graphs

“…G δ≥2 n,m is not Hamiltonian. On the other hand Anastos and Frieze prove that taking k = 3 and m ≥ 2.67n suffices [3].…”