5 research outputs found
Long induced paths in expanders
We prove that any bounded degree regular graph with sufficiently strong
spectral expansion contains an induced path of linear length. This is the first
such result for expanders, strengthening an analogous result in the random
setting by Dragani\'c, Glock and Krivelevich. More generally, we find long
induced paths in sparse graphs that satisfy a mild upper-uniformity
edge-distribution condition.Comment: 7 page
Short proofs for long induced paths
We present a modification of the DFS graph search algorithm, suited for
finding long induced paths. We use it to give simple proofs of the following
results. We show that the induced size-Ramsey number of paths satisfies
, thus giving an explicit
constant in the linear bound, improving the previous bound with a large
constant from a regularity lemma argument by Haxell, Kohayakawa and {\L}uczak.
We also provide a bound for the -color version, showing that
. Finally, we present a new short
proof of the fact that the binomial random graph in the supercritical regime,
, contains typically an induced path of length
.Comment: 9 page
Large induced matchings in random graphs
Given a large graph , does the binomial random graph contain a
copy of as an induced subgraph with high probability? This classical
question has been studied extensively for various graphs , going back to the
study of the independence number of by Erd\H{o}s and Bollob\'as, and
Matula in 1976. In this paper we prove an asymptotically best possible result
for induced matchings by showing that if for some large
constant , then contains an induced matching of order approximately
, where