scholarly journals Permutation Pseudographs and Contiguity

2002 ◽  
Vol 11 (3) ◽  
pp. 273-298 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
JEONG HAN KIM ◽  
NICHOLAS C. WORMALD
Keyword(s):  
Regular Graph ◽  
Probabilistic Model ◽  
General Setting ◽  
Hamilton Cycle ◽  
Asymptotic Equivalence ◽  
Arbitrary Degree ◽  
Simple Graphs ◽  
Nonnegative Constant ◽  
Random Regular Graph ◽  
Probability Spaces

The space of permutation pseudographs is a probabilistic model of 2-regular pseudographs on n vertices, where a pseudograph is produced by choosing a permutation σ of {1,2,…, n} uniformly at random and taking the n edges {i,σ(i)}. We prove several contiguity results involving permutation pseudographs (contiguity is a kind of asymptotic equivalence of sequences of probability spaces). Namely, we show that a random 4-regular pseudograph is contiguous with the sum of two permutation pseudographs, the sum of a permutation pseudograph and a random Hamilton cycle, and the sum of a permutation pseudograph and a random 2-regular pseudograph. (The sum of two random pseudograph spaces is defined by choosing a pseudograph from each space independently and taking the union of the edges of the two pseudographs.) All these results are proved simultaneously by working in a general setting, where each cycle of the permutation is given a nonnegative constant multiplicative weight. A further contiguity result is proved involving the union of a weighted permutation pseudograph and a random regular graph of arbitrary degree. All corresponding results for simple graphs are obtained as corollaries.

SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

2016 ◽  
Vol 59 (3) ◽  
pp. 533-547 ◽  
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
General Setting ◽  
Maximal Operators ◽  
List Type ◽  
Type Number ◽  
Type Estimate ◽  
Formula Group ◽  
Probability Spaces ◽  
Image Position

AbstractLet $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$. (i)We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$ For each p, the constant e1/p is the best possible.(ii)We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$ and prove that the constant e is optimal. Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

scholarly journals Optimal Construction of Edge-Disjoint Paths in Random Regular Graphs

2000 ◽  
Vol 9 (3) ◽  
pp. 241-263 ◽  
Author(s):  
ALAN M. FRIEZE ◽  
LEI ZHAO
Keyword(s):  
Polynomial Time ◽  
Regular Graph ◽  
Randomized Algorithm ◽  
Constant Factor ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Edge Disjoint ◽  
Random Regular Graph ◽  
Arbitrary Graphs ◽  
Optimal Construction

Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is [Nscr ][Pscr ]-complete, although it is in [Pscr ] if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.

scholarly journals Expansion properties of a random regular graph after random vertex deletions

2008 ◽  
Vol 29 (5) ◽  
pp. 1139-1150 ◽  
Author(s):  
Catherine Greenhill ◽  
Fred B. Holt ◽  
Nicholas Wormald
Keyword(s):  
Regular Graph ◽  
Random Regular Graph

scholarly journals Graph Products and New Solutions to Oberwolfach Problems

10.37236/539 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Gloria Rinaldi ◽  
Tommaso Traetta
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Graph Products ◽  
Simple Graphs ◽  
Oberwolfach Problem

We introduce the circle product, a method to construct simple graphs starting from known ones. The circle product can be applied in many different situations and when applied to regular graphs and to their decompositions, a new regular graph is obtained together with a new decomposition. In this paper we show how it can be used to construct infinitely many new solutions to the Oberwolfach problem, in both the classic and the equipartite case.

Induced Subgraph in Random Regular Graph

2008 ◽  
Vol 21 (4) ◽  
pp. 645-650
Author(s):  
Lan XIAO ◽  
Guiying YAN ◽  
Yuwen WU ◽  
Wei REN
Keyword(s):  
Regular Graph ◽  
Induced Subgraph ◽  
Random Regular Graph

scholarly journals Dismantling Sparse Random Graphs

2008 ◽  
Vol 17 (2) ◽  
pp. 259-264 ◽  
Author(s):  
SVANTE JANSON ◽  
ANDREW THOMASON
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Regular Graph ◽  
Random Regular Graph

We consider the number of vertices that must be removed from a graphGin order that the remaining subgraph have no component with more thankvertices. Our principal observation is that, ifGis a sparse random graph or a random regular graph onnvertices withn→ ∞, then the number in question is essentially the same for all values ofkthat satisfy bothk→ ∞ andk=o(n).

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