We develop a general embedding method based on the Friedman-Pippenger tree embedding technique and its algorithmic version, enhanced with a roll-back idea allowing a sequential retracing of previously performed embedding steps. We use this method to obtain the following results.
We show that the size-Ramsey number of logarithmically long subdivisions of bounded degree graphs is linear in their number of vertices, settling a conjecture of Pak [Proceedings of the thirteenth annual ACMSIAM symposium on discrete algorithms (SODA’02), 2002, pp. 321-328].
We give a deterministic, polynomial time online algorithm for finding vertex-disjoint paths of a prescribed length between given pairs of vertices in an expander graph. Our result answers a question of Alon and Capalbo [48th annual IEEE symposium on foundations of computer science (FOCS’07), 2007, pp. 518-524].
We show that relatively weak bounds on the spectral ratio
λ
/
d
\lambda /d
of
d
d
-regular graphs force the existence of a topological minor of
K
t
K_t
where
t
=
(
1
−
o
(
1
)
)
d
t=(1-o(1))d
. We also exhibit a construction which shows that the theoretical maximum
t
=
d
+
1
t=d+1
cannot be attained even if
λ
=
O
(
d
)
\lambda =O(\sqrt {d})
. This answers a question of Fountoulakis, Kühn and Osthus [Random Structures Algorithms 35 (2009), pp. 444-463].