153,342 research outputs found
A bandwidth theorem for approximate decompositions
We provide a degree condition on a regular -vertex graph which ensures
the existence of a near optimal packing of any family of bounded
degree -vertex -chromatic separable graphs into . In general, this
degree condition is best possible.
Here a graph is separable if it has a sublinear separator whose removal
results in a set of components of sublinear size. Equivalently, the
separability condition can be replaced by that of having small bandwidth. Thus
our result can be viewed as a version of the bandwidth theorem of B\"ottcher,
Schacht and Taraz in the setting of approximate decompositions.
More precisely, let be the infimum over all
ensuring an approximate -decomposition of any sufficiently large regular
-vertex graph of degree at least . Now suppose that is an
-vertex graph which is close to -regular for some and suppose that is a sequence of bounded
degree -vertex -chromatic separable graphs with . We show that there is an edge-disjoint packing of
into .
If the are bipartite, then is sufficient. In
particular, this yields an approximate version of the tree packing conjecture
in the setting of regular host graphs of high degree. Similarly, our result
implies approximate versions of the Oberwolfach problem, the Alspach problem
and the existence of resolvable designs in the setting of regular host graphs
of high degree.Comment: Final version, to appear in the Proceedings of the London
Mathematical Societ
Unavoidable minors in graphs and matroids
It is well known that every sufficiently large connected graph G has either a vertex of high degree or a long path. If we require G to be more highly connected, then we ensure the presence of more highly structured minors. In particular, for all positive integers k, every 2-connected graph G has a series minor isomorphic to a k-edge cycle or K_{2,k}. In 1993, Oxley, Oporowski, and Thomas extended this result to 3- and internally 4-connected graphs identifying all unavoidable series minors of these classes. Loosely speaking, a series minor allows for arbitrary edge deletions but only allows edges to be contracted when they meet a degree-2 vertex. Dually, a parallel minor allows for any edge contractions but restricts the deletion of edges to those that lie in 2-edge cycles. This dissertation begins by proving the dual results to those noted above. These identify all unavoidable parallel minors for finite graphs of low connectivity. Following this, corresponding results on unavoidable minors for infinite graphs are proved. The dissertation concludes by finding the unavoidable parallel minors for 3-connected regular matroids, which combines the results for unavoidable series and parallel minors for graphs with Seymour\u27s decomposition theorem for regular matroids
Random walks which prefer unvisited edges : exploring high girth even degree expanders in linear time.
Let G = (V,E) be a connected graph with |V | = n vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge-process (or E -process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E -process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap 1 -λmax of the transition matrix of a simple random walk on G. A vertex v is ℓ -good, if any even degree subgraph containing all edges incident with v contains at least ℓ vertices. A graph G is ℓ -good, if every vertex has the ℓ -good property. Let G be an even degree ℓ -good expander of bounded maximum degree. Any E -process on G has vertex cover time
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This is to be compared with the Ω(nlog n) lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on-line by an adversary. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2013
As no walk based process can cover an n vertex graph in less than n - 1 steps, the cover time of the E -process is of optimal order when ℓ =Θ (log n). With high probability random r -regular graphs, r ≥ 4 even, have ℓ =Ω (log n). Thus the vertex cover time of the E -process on such graphs is Θ(n)
Universal lower bound for community structure of sparse graphs
We prove new lower bounds on the modularity of graphs. Specifically, the
modularity of a graph with average degree is
, under some mild assumptions on the degree sequence of
. The lower bound applies, for instance, to graphs
with a power-law degree sequence or a near-regular degree sequence.
It has been suggested that the relatively high modularity of the
Erd\H{o}s-R\'enyi random graph stems from the random fluctuations in
its edge distribution, however our results imply high modularity for any graph
with a degree sequence matching that typically found in .
The proof of the new lower bound relies on certain weight-balanced bisections
with few cross-edges, which build on ideas of Alon [Combinatorics, Probability
and Computing (1997)] and may be of independent interest.Comment: 25 pages, 2 figure
Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs
For an increasing monotone graph property \mP the \emph{local resilience}
of a graph with respect to \mP is the minimal for which there exists
of a subgraph with all degrees at most such that the removal
of the edges of from creates a graph that does not possesses \mP.
This notion, which was implicitly studied for some ad-hoc properties, was
recently treated in a more systematic way in a paper by Sudakov and Vu. Most
research conducted with respect to this distance notion focused on the Binomial
random graph model \GNP and some families of pseudo-random graphs with
respect to several graph properties such as containing a perfect matching and
being Hamiltonian, to name a few. In this paper we continue to explore the
local resilience notion, but turn our attention to random and pseudo-random
\emph{regular} graphs of constant degree. We investigate the local resilience
of the typical random -regular graph with respect to edge and vertex
connectivity, containing a perfect matching, and being Hamiltonian. In
particular we prove that for every positive and large enough values
of with high probability the local resilience of the random -regular
graph, \GND, with respect to being Hamiltonian is at least .
We also prove that for the Binomial random graph model \GNP, for every
positive and large enough values of , if
then with high probability the local resilience of \GNP with respect to being
Hamiltonian is at least . Finally, we apply similar
techniques to Positional Games and prove that if is large enough then with
high probability a typical random -regular graph is such that in the
unbiased Maker-Breaker game played on the edges of , Maker has a winning
strategy to create a Hamilton cycle.Comment: 34 pages. 1 figur
Packing and embedding large subgraphs
This thesis contains several embedding results for graphs in both random and non random settings.
Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals . %posed e.g.~by Bollob\'as,
In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n:
if H\subseteq\cQ^n satisfies with fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with fixed, then with high probability H\cup\cQ^n_p contains edge-disjoint Hamilton cycles, for any fixed .
This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity.
In Chapter 3 we move to a non random setting. %to a deterministic one.
%Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph.
Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs.
More specifically, we provide a degree condition on a regular -vertex graph which ensures the existence of a near optimal packing of any family of bounded degree -vertex -chromatic separable graphs into .
%In general, this degree condition is best possible.
%In particular, this yields an approximate version of the tree packing conjecture
%in the setting of regular host graphs of high degree.
%Similarly, our result implies approximate versions of the Oberwolfach problem,
%the Alspach problem and the existence of resolvable designs in the setting of
%regular host graphs of high degree.
In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem,
the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree
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