550 research outputs found
Local resilience of spanning subgraphs in sparse random graphs
For each real γ>0γ>0 and integers Δ≥2Δ≥2 and k≥1k≥1, we prove that there exist constants β>0β>0 and C>0C>0 such that for all p≥C(logn/n)1/Δp≥C(logn/n)1/Δ the random graph G(n,p)G(n,p) asymptotically almost surely contains – even after an adversary deletes an arbitrary (1/k−γ1/k−γ)-fraction of the edges at every vertex – a copy of every n-vertex graph with maximum degree at most Δ, bandwidth at most βn and at least Cmax{p−2,p−1logn}Cmax{p−2,p−1logn} vertices not in triangles
Almost spanning subgraphs of random graphs after adversarial edge removal
Let Delta>1 be a fixed integer. We show that the random graph G(n,p) with
p>>(log n/n)^{1/Delta} is robust with respect to the containment of almost
spanning bipartite graphs H with maximum degree Delta and sublinear bandwidth
in the following sense: asymptotically almost surely, if an adversary deletes
arbitrary edges in G(n,p) such that each vertex loses less than half of its
neighbours, then the resulting graph still contains a copy of all such H.Comment: 46 pages, 6 figure
Local resilience for squares of almost spanning cycles in sparse random graphs
In 1962, P\'osa conjectured that a graph contains a square of a
Hamiltonian cycle if . Only more than thirty years later
Koml\'os, S\'ark\H{o}zy, and Szemer\'edi proved this conjecture using the
so-called Blow-Up Lemma. Here we extend their result to a random graph setting.
We show that for every and a.a.s. every
subgraph of with minimum degree at least contains
the square of a cycle on vertices. This is almost best possible in
three ways: (1) for the random graph will not contain any
square of a long cycle (2) one cannot hope for a resilience version for the
square of a spanning cycle (as deleting all edges in the neighborhood of single
vertex destroys this property) and (3) for a.a.s. contains a
subgraph with minimum degree at least which does not contain the square
of a path on vertices
Dirac's theorem for random regular graphs
We prove a `resilience' version of Dirac's theorem in the setting of random
regular graphs. More precisely, we show that, whenever is sufficiently
large compared to , a.a.s. the following holds: let be any
subgraph of the random -vertex -regular graph with minimum
degree at least . Then is Hamiltonian.
This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result
is best possible: firstly, the condition that is large cannot be omitted,
and secondly, the minimum degree bound cannot be improved.Comment: Final accepted version, to appear in Combinatorics, Probability &
Computin
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
Generating random graphs in biased Maker-Breaker games
We present a general approach connecting biased Maker-Breaker games and
problems about local resilience in random graphs. We utilize this approach to
prove new results and also to derive some known results about biased
Maker-Breaker games. In particular, we show that for
, Maker can build a pancyclic graph (that is, a graph
that contains cycles of every possible length) while playing a game on
. As another application, we show that for , playing a game on , Maker can build a graph which
contains copies of all spanning trees having maximum degree with
a bare path of linear length (a bare path in a tree is a path with all
interior vertices of degree exactly two in )
Bandwidth theorem for random graphs
A graph is said to have \textit{bandwidth} at most , if there exists a
labeling of the vertices by , so that whenever
is an edge of . Recently, B\"{o}ttcher, Schacht, and Taraz
verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every
positive , there exists such that if is an
-vertex -chromatic graph with maximum degree at most which has
bandwidth at most , then any graph on vertices with minimum
degree at least contains a copy of for large enough
. In this paper, we extend this theorem to dense random graphs. For
bipartite , this answers an open question of B\"{o}ttcher, Kohayakawa, and
Taraz. It appears that for non-bipartite the direct extension is not
possible, and one needs in addition that some vertices of have independent
neighborhoods. We also obtain an asymptotically tight bound for the maximum
number of vertex disjoint copies of a fixed -chromatic graph which one
can find in a spanning subgraph of with minimum degree .Comment: 29 pages, 3 figure
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