20 research outputs found
Local resilience of an almost spanning -cycle in random graphs
The famous P\'{o}sa-Seymour conjecture, confirmed in 1998 by Koml\'{o}s,
S\'{a}rk\"{o}zy, and Szemer\'{e}di, states that for any , every graph
on vertices with minimum degree contains the -th power of a
Hamilton cycle. We extend this result to a sparse random setting.
We show that for every there exists such that if then w.h.p. every subgraph of a random graph with
minimum degree at least , contains the -th power of a
cycle on at least vertices, improving upon the recent results of
Noever and Steger for , as well as Allen et al. for .
Our result is almost best possible in three ways: for the
random graph w.h.p. does not contain the -th power of any long
cycle; there exist subgraphs of with minimum degree and vertices not belonging to triangles; there exist
subgraphs of with minimum degree which do not
contain the -th power of a cycle on vertices.Comment: 24 pages; small updates to the paper after anonymous reviewers'
report
Monochromatic cycle covers in random graphs
A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every
coloring of the edges of with colors, there is a cover of its vertex
set by at most vertex-disjoint monochromatic cycles. In
particular, the minimum number of such covering cycles does not depend on the
size of but only on the number of colors. We initiate the study of this
phenomena in the case where is replaced by the random graph . Given a fixed integer and , we
show that with high probability the random graph has
the property that for every -coloring of the edges of , there is a
collection of monochromatic cycles covering all the
vertices of . Our bound on is close to optimal in the following sense:
if , then with high probability there are colorings of
such that the number of monochromatic cycles needed to
cover all vertices of grows with .Comment: 24 pages, 1 figure (minor changes, added figure
Almost-spanning universality in random graphs (Extended abstract)
A graph G is said to be ℋ(n, Δ)-universal if it contains every graph on n vertices with maximum degree at most Δ. It is known that for any ε > 0 and any natural number Δ there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely ℋ((1 - ε)n, Δ)-universal for p ≥ c(log n/n)^(1/Δ). Bypassing this natural boundary Δ ≥ 3, we show that for the same conclusion holds when [equation; see abstract in PDF for details]