Rainbow Matchings and Hamilton Cycles in Random Graphs

Let $HP_{n,m,k}$ be drawn uniformly from all $k$-uniform, $k$-partite hypergraphs where each part of the partition is a disjoint copy of $[n]$. We let HP^{(\k)}_{n,m,k} be an edge colored version, where we color each edge randomly from one of \k colors. We show that if \k=n and $m=Kn\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if $n$ is even and $m=Kn\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in $G^{(n)}_{n,m}$. Here $G^{(n)}_{n,m}$ denotes a random edge coloring of $G_{n,m}$ with $n$ colors. When $n$ is odd, our proof requires m=\om(n\log n) for there to be a rainbow Hamilton cycle.Comment: We replaced graphs by k-uniform hypergraph

Optimal covers with Hamilton cycles in random graphs

A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szab\'o 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 G_n,p can be covered by \lceil Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szab\'o, which holds for p > n^{-1+\eps}. Our proof is based on a result of Knox, K\"uhn and Osthus on packing Hamilton cycles in pseudorandom graphs.Comment: final version of paper (to appear in Combinatorica

Pancyclic Hamilton cycles in random graphs

AbstractLet G(n,p) denote the probability space of the set G of graphs G = (Vn, E) with vertex set Vn = {1,2,…, n} and edges E chosen independently with probability p from E={{u,v}:u,v∈Vn,u≠v}.A graph G∈G(n,p is defined to be pancyclic if, for all s, 3⩽s⩽n there is a cycle of size s on the edges of G. We show that the threshold probability p = (log n + log log n + cn)/n for the property that G contains a Hamilton cycle is also the threshold probability for the existence of a 2-pancyclic Hamilton cycle, which is defined as follows. Given a Hamilton cycle H, we will say that H is k-pancyclic if for each s (3⩽s⩽n−1) we can find a cycle C of length s using only the edges of H and at most k other edges

Colorful Hamilton cycles in random graphs

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C=\{c_1,c_2,\ldots,c_r\}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,\ldots,m_r)\in [0,n]^r$ such that there exists a Hamilton cycle that is the concatenation of $r$ paths $P_1,P_2,\ldots,P_r$, where $P_i$ contains $m_i$ edges. We study $hcp(G_{n,p})$ when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when $hcp(G_{n,p})=\{(m_1,m_2,\ldots,m_r)\in [0,n]^r:m_1+m_2+\cdots+m_r=n\}$.Comment: minor changes reflecting comments from an anonymous refere