On Perfect Matchings and Hamilton Cycles in Sums of Random Trees

The Numbers of Spanning Trees, Hamilton cycles and Perfect Matchings in a Random Graph

Expected numbers at hitting times

We determine exactly the expected number of hamilton cycles in the random graph obtained by starting with n isolated vertices and adding edges at random until each vertex degree is at least two. This complements recent work of Cooper and Frieze. There are similar results concerning expected numbers for example of perfect matchings, spanning trees, hamilton paths and directed hamilton cycles.

Random Matchings Which Induce Hamilton Cycles and Hamiltonian Decompositions of Random Regular Graphs

Select four perfect matchings of 2n vertices, independently at random. We find the asymptotic probability that each of the first and second matchings forms a Hamilton cycle with each of the third and fourth. This is generalised to embrace any fixed number of perfect matchings, where a prescribed set of pairs of matchings must each produce Hamilton cycles (with suitable restrictions on the presc...

Multicoloured Hamilton cycles and perfect matchings in pseudo-random graphs

Given 0 < p < 1, we prove that a pseudo-random graph G with edge density p and sufficiently large order has the following property: Consider any red/blue-colouring of the edges of G and let r denote the proportion of edges which have colour red. Then there is a Hamilton cycle C so that the proportion of red edges of C is close to r. The analogue also holds for perfect matchings instead of Hamil...

On Perfect Matchings and Hamiltonian Cycles in Sumsof Random

We prove that the sum of two random trees almost surely possesses a perfect matching and the sum of ve random trees almost surely possesses a Hamiltonian cycle.