10.
Given an
‐vertex pseudorandom graph
and an
‐vertex graph
with maximum degree at most two, we wish to find a copy of
in
, that is, an embedding
so that
for all
. Particular instances of this problem include finding a triangle‐factor and finding a Hamilton cycle in
. Here, we provide a deterministic polynomial time algorithm that finds a given
in any suitably pseudorandom graph
. The pseudorandom graphs we consider are
‐bijumbled graphs of minimum degree which is a constant proportion of the average degree, that is,
. A
‐bijumbled graph is characterised through the discrepancy property:
for any two sets of vertices
and
. Our condition
on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption‐reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work, incorporating the work of Nenadov, together with additional ideas and simplifications.
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