5,192 research outputs found
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 )
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
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 heuristics and the emergence of spanning subgraphs in complex networks
We study the use of local heuristics to determine spanning subgraphs for use
in the dissemination of information in complex networks. We introduce two
different heuristics and analyze their behavior in giving rise to spanning
subgraphs that perform well in terms of allowing every node of the network to
be reached, of requiring relatively few messages and small node bandwidth for
information dissemination, and also of stretching paths with respect to the
underlying network only modestly. We contribute a detailed mathematical
analysis of one of the heuristics and provide extensive simulation results on
random graphs for both of them. These results indicate that, within certain
limits, spanning subgraphs are indeed expected to emerge that perform well in
respect to all requirements. We also discuss the spanning subgraphs' inherent
resilience to failures and adaptability to topological changes
Embedding bounded degree spanning trees in random graphs
We prove that if a tree has vertices and maximum degree at most
, then a copy of can almost surely be found in the random graph
.Comment: 14 page
- …