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 $$a = 2$$