[HTML][HTML] Hamilton cycles in random lifts of graphs

K Burgin, P Chebolu, C Cooper, AM Frieze - European Journal of …, 2006 - Elsevier
An n-lift of a graph K is a graph with vertex set V (K)×[n], and for each edge (i, j)∈ E (K) there
is a perfect matching between {i}×[n] and {j}×[n]. If these matchings are chosen
independently and uniformly at random then we say that we have a random n-lift. We show
that there are constants h1, h2 such that if h≥ h1 then a random n-lift of the complete graph
Kh is hamiltonian whp and if h≥ h2 then a random n-lift of the complete bipartite graph Kh, h
is hamiltonian whp.

[HTML][HTML] Hamilton cycles in random lifts of graphs

T Łuczak, Ł Witkowski, M Witkowski - European Journal of Combinatorics, 2015 - Elsevier
For a graph G the random n-lift of G is obtained by replacing each of its vertices by a set of n
vertices, and joining a pair of sets by a random matching whenever the corresponding
vertices of G are adjacent. We show that asymptotically almost surely the random lift of a
graph G is Hamiltonian, provided G has the minimum degree at least 5 and contains two
disjoint Hamiltonian cycles whose union is not a bipartite graph.
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