26,273 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Hitting time results for Maker-Breaker games
We study Maker-Breaker games played on the edge set of a random graph.
Specifically, we consider the random graph process and analyze the first time
in a typical random graph process that Maker starts having a winning strategy
for his final graph to admit some property \mP. We focus on three natural
properties for Maker's graph, namely being -vertex-connected, admitting a
perfect matching, and being Hamiltonian. We prove the following optimal hitting
time results: with high probability Maker wins the -vertex connectivity game
exactly at the time the random graph process first reaches minimum degree ;
with high probability Maker wins the perfect matching game exactly at the time
the random graph process first reaches minimum degree ; with high
probability Maker wins the Hamiltonicity game exactly at the time the random
graph process first reaches minimum degree . The latter two statements
settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page
Freezing in random graph ferromagnets
Using T=0 Monte Carlo and simulated annealing simulation, we study the energy
relaxation of ferromagnetic Ising and Potts models on random graphs. In
addition to the expected exponential decay to a zero energy ground state, a
range of connectivities for which there is power law relaxation and freezing to
a metastable state is found. For some connectivities this freezing persists
even using simulated annealing to find the ground state. The freezing is caused
by dynamic frustration in the graphs, and is a feature of the local
search-nature of the Monte Carlo dynamics used. The implications of the
freezing on agent-based complex systems models are briefly considered.Comment: Published version: 1 reference deleted, 1 word added. 4 pages, 5
figure
All reducts of the random graph are model-complete
We study locally closed transformation monoids which contain the automorphism
group of the random graph. We show that such a transformation monoid is locally
generated by the permutations in the monoid, or contains a constant operation,
or contains an operation that maps the random graph injectively to an induced
subgraph which is a clique or an independent set. As a corollary, our
techniques yield a new proof of Simon Thomas' classification of the five closed
supergroups of the automorphism group of the random graph; our proof uses
different Ramsey-theoretic tools than the one given by Thomas, and is perhaps
more straightforward. Since the monoids under consideration are endomorphism
monoids of relational structures definable in the random graph, we are able to
draw several model-theoretic corollaries: One consequence of our result is that
all structures with a first-order definition in the random graph are
model-complete. Moreover, we obtain a classification of these structures up to
existential interdefinability.Comment: Technical report not intended for publication in a journal. Subsumed
by the more recent article 1003.4030. Length 14 pages
Random Graph-Homomorphisms and Logarithmic Degree
A graph homomorphism between two graphs is a map from the vertex set of one
graph to the vertex set of the other graph, that maps edges to edges. In this
note we study the range of a uniformly chosen homomorphism from a graph G to
the infinite line Z. It is shown that if the maximal degree of G is
`sub-logarithmic', then the range of such a homomorphism is super-constant.
Furthermore, some examples are provided, suggesting that perhaps for graphs
with super-logarithmic degree, the range of a typical homomorphism is bounded.
In particular, a sharp transition is shown for a specific family of graphs
C_{n,k} (which is the tensor product of the n-cycle and a complete graph, with
self-loops, of size k). That is, given any function psi(n) tending to infinity,
the range of a typical homomorphism of C_{n,k} is super-constant for k = 2
log(n) - psi(n), and is 3 for k = 2 log(n) + psi(n)
- …