17 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
On resilience of connectivity in the evolution of random graphs
In this note we establish a resilience version of the classical hitting time
result of Bollob\'{a}s and Thomason regarding connectivity. A graph is said
to be -resilient with respect to a monotone increasing graph property
if for every spanning subgraph satisfying
for all ,
the graph still possesses . Let be the random
graph process, that is a process where, starting with an empty graph on
vertices , in each step an edge is chosen uniformly at
random among the missing ones and added to the graph . We show that
the random graph process is almost surely such that starting from , the largest connected component of is
-resilient with respect to connectivity. The result is
optimal in the sense that the constants in the number of edges and
in the resilience cannot be improved upon. We obtain similar results for
-connectivity.Comment: 13 pages; update after reviewers' report
Ramsey numbers for multiple copies of sparse graphs
For a graph H and an integer n, we let nH denote the disjoint union of n copies of H. In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for nH, one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant c = c(H) such that r(nH) = (2 | H | - α (H))n + c, provided n is sufficiently large. Subsequently, Burr gave an implicit way of computing c and noted that this long-term behaviour occurs when n is triply exponential in | H |. Very recently, Bucić and Sudakov revived the problem and established an essentially tight bound on n by showing r(nH) follows this behaviour already when the number of copies is just a single exponential. We provide significantly stronger bounds on n in case H is a sparse graph, most notably of bounded maximum degree. These are relatable to the current state-of-the-art bounds on r(H) and (in a way) tight. Our methods rely on a beautiful classic proof of Graham, Rödl and Ruciński, with an emphasis on developing an efficient absorbing method for bounded degree graphs.ISSN:0364-9024ISSN:1097-011