Rainbow Hamilton cycles in random graphs and hypergraphs

Abstract

Let H be an edge colored hypergraph. We say that H contains a rainbow copy of a hypergraph S if it contains an isomorphic copy of S with all edges of distinct colors.We consider the following setting. A randomly edge colored random hypergraph H ∼ H k c (n, p) is obtained by adding each k-subset of [n] with probability p, and assigning it a color from [c] uniformly, independently at random.As a first result we show that a typical H ∼ H 2 c (n, p) (that is, a random edge colored graph) contains a rainbow Hamilton cycle, provided that c = (1 + o(1))n and p = log n+log log n+ω (1) n . This is asymptotically best possible with respect to both parameters, and improves a result of Frieze and Loh.Secondly, based on an ingenious coupling idea of McDiarmid, we provide a general tool for tackling problems related to finding "nicely edge colored" structures in random graphs/hypergraphs. We illustrate the generality of this statement by presenting two interesting applications. In one application we show that a typical H ∼ H k c (n, p) contains a rainbow copy of a hypergraph S, provided that c = (1 + o(1))|E(S)| and p is (up to a multiplicative constant) a threshold function for the property of containment of a copy of S. In the second application we show that a typical G ∼ H 2 c (n, p) contains (1 − o(1))np/2 edge disjoint Hamilton cycles, each of which is rainbow, provided that c = ω(n) and p = ω(log n/n).

Pattern Colored Hamilton Cycles in Random Graphs

“…This was improved to (1 + o(1))n log n random edges and (1 + o(1))n colors by Frieze and Loh [9]. This was sharpened still further by Ferber and Krivelevich [7] who showed that the number of edges can be reduced to the exact threshold for Hamiltonicity. Bal and Frieze considered the case where exactly n colors are available and showed that O(n log n) random edges are sufficient to obtain a rainbow Hamilton cycle.…”

“…We also prove a hitting time result for a related notion of Π-connected.The next phase of this study, concerns Hamilton cycles in random k-uniform hypergraphs. There are various notions of Hamilton cycle in this context, and Ferber and Krivelevich [7] proved that if m H edges are needed for a given type of hamilton cycle to exist w.h.p. then O(m H ) random edges and (1 + ǫ)m H colors are sufficient for a rainbow Hamilton cycles.…”

“…then O(m H ) random edges and (1 + ǫ)m H colors are sufficient for a rainbow Hamilton cycles. The hypergraph results in [7] were sharpened by Dudek, English and Frieze [5]. Cooper and Frieze [4] considered the related question of what is the threshold for every k-bounded coloring of the edges of G n,m to contain at least one rainbow Hamilton cycle.Here k-bounded means that no color can be used more than k times.…”

How many randomly colored edges make a randomly colored dense graph rainbow Hamiltonian or rainbow connected?

“…A Hamilton cycle is called rainbow if no color appears twice on its edges. It was shown by Frieze and Loh and by Ferber and Krivelevich that for , if we color randomly with colors, then contains a rainbow‐Hamilton cycle. This implies that if we randomly color a typical graph of minimum degree , then w.h.p.…”

Counting Hamilton cycles in sparse random directed graphs

“…One of the first important insights concerning Hamilton cycles in random digraphs, due to McDiarmid , is that one can use coupling arguments to compare certain probabilities between and the random undirected graph (see for more recent applications of McDiarmid's idea). In particular, using the known optimal results for , McDiarmid's work implies that if then a.a.s .…”