Introduction
Given a random graph which can be constructed as follows:
generate $s$ pairs of unconnected black edges and $2 \cdot s$ vertices
connect these black edges using red edges, where the average number of red edges on a vertex is $a$
Question
What is the chance that, starting from a given vertex and following a random walk of alternating black and red edges that one doesn't end up with a Hamiltonian path in $n$ tries (in terms of $n$, $s$ and $a$). Or a good upperbound
Also is there a difference between this chance for a given $s$ and a graph with $s+k\text{ }$ black edges where one starts the walk with $k$ black edges already visited
Example:
a graph where $s = 3$ and $a = 2$