13 research outputs found
3-star factors in random d-regular graphs
AbstractThe small subgraph conditioning method first appeared when Robinson and the second author showed the almost sure hamiltonicity of random d-regular graphs. Since then it has been used to study the almost sure existence of, and the asymptotic distribution of, regular spanning subgraphs of various types in random d-regular graphs and hypergraphs. In this paper, we use the method to prove the almost sure existence of 3-star factors in random d-regular graphs. This is essentially the first application of the method to non-regular subgraphs in such graphs
Approximately Counting Embeddings into Random Graphs
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic)
copies of H contained in a graph G. We investigate the fundamental problem of
estimating C_H(G). Previous results cover only a few specific instances of this
general problem, for example, the case when H has degree at most one
(monomer-dimer problem). In this paper, we present the first general subcase of
the subgraph isomorphism counting problem which is almost always efficiently
approximable. The results rely on a new graph decomposition technique.
Informally, the decomposition is a labeling of the vertices such that every
edge is between vertices with different labels and for every vertex all
neighbors with a higher label have identical labels. The labeling implicitly
generates a sequence of bipartite graphs which permits us to break the problem
of counting embeddings of large subgraphs into that of counting embeddings of
small subgraphs. Using this method, we present a simple randomized algorithm
for the counting problem. For all decomposable graphs H and all graphs G, the
algorithm is an unbiased estimator. Furthermore, for all graphs H having a
decomposition where each of the bipartite graphs generated is small and almost
all graphs G, the algorithm is a fully polynomial randomized approximation
scheme.
We show that the graph classes of H for which we obtain a fully polynomial
randomized approximation scheme for almost all G includes graphs of degree at
most two, bounded-degree forests, bounded-length grid graphs, subdivision of
bounded-degree graphs, and major subclasses of outerplanar graphs,
series-parallel graphs and planar graphs, whereas unbounded-length grid graphs
are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition
3.
Cycle factors and renewal theory
For which values of does a uniformly chosen -regular graph on
vertices typically contain vertex-disjoint -cycles (a -cycle
factor)? To date, this has been answered for and for ; the
former, the Hamiltonicity problem, was finally answered in the affirmative by
Robinson and Wormald in 1992, while the answer in the latter case is negative
since with high probability most vertices do not lie on -cycles.
Here we settle the problem completely: the threshold for a -cycle factor
in as above is with . Precisely, we prove a 2-point concentration result: if divides then contains a -cycle factor
w.h.p., whereas if then w.h.p. it
does not. As a byproduct, we confirm the "Comb Conjecture," an old problem
concerning the embedding of certain spanning trees in the random graph
.
The proof follows the small subgraph conditioning framework, but the
associated second moment analysis here is far more delicate than in any earlier
use of this method and involves several novel features, among them a sharp
estimate for tail probabilities in renewal processes without replacement which
may be of independent interest.Comment: 45 page
Quantitative Small Subgraph Conditioning
We revisit the method of small subgraph conditioning, used to establish that
random regular graphs are Hamiltonian a.a.s. We refine this method using new
technical machinery for random -regular graphs on vertices that hold not
just asymptotically, but for any values of and . This lets us estimate
how quickly the probability of containing a Hamiltonian cycle converges to 1,
and it produces quantitative contiguity results between different models of
random regular graphs. These results hold with held fixed or growing to
infinity with . As additional applications, we establish the distributional
convergence of the number of Hamiltonian cycles when grows slowly to
infinity, and we prove that the number of Hamiltonian cycles can be
approximately computed from the graph's eigenvalues for almost all regular
graphs.Comment: 59 pages, 5 figures; minor changes for clarit
Generating and counting Hamilton cycles in random regular graphs
SIGLEAvailable from British Library Document Supply Centre- DSC:5186.0913(EU-ECS-LFCS--94-313) / BLDSC - British Library Document Supply CentreGBUnited Kingdo