zufällige graphen - uni ulm...juni 2010 | zhibin huang in graph theory, the girth of a graph is the...
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Zufällige Graphen
Zhibin Huang | 07. Juni 2010
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Zufällige Graphen | 07. Juni 2010 | Zhibin Huang Seite 2
Contents
The Basic Method
The Probabilistic Method
The Ramsey Number
Linearity of Expectation
Basics
Splitting Graphs
The Probabilistic Lens:
High Girth and High Chromatic Number
( , )R k l
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Contents
The Basic Method
The Probabilistic Method
The Ramsey Number
Linearity of Expectation
Basics
Splitting Graphs
The Probabilistic Lens:
High Girth and High Chromatic Number
( , )R k l
![Page 4: Zufällige Graphen - Uni Ulm...Juni 2010 | Zhibin Huang In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph.If the graph does not contain](https://reader036.vdocument.in/reader036/viewer/2022081622/613bc368f8f21c0c82692e60/html5/thumbnails/4.jpg)
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Trying to prove that a structure with certain desired properties
exists, one defines an appropriate probility space of structures
and then shows that the desired properties hold in this space
with positive probability.
The Probabilistic Method
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Content
The Basic Method
The Probabilistic Method
The Ramsey Number
Linearity of Expectation
Basics
Splitting Graphs
The Probabilistic Lens:
High Girth and High Chromatic Number
( , )R k l
![Page 6: Zufällige Graphen - Uni Ulm...Juni 2010 | Zhibin Huang In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph.If the graph does not contain](https://reader036.vdocument.in/reader036/viewer/2022081622/613bc368f8f21c0c82692e60/html5/thumbnails/6.jpg)
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A complete graph is a simple graph in which every pair of distinct
vertices is connected by a unique edge.
The complete graph on n vertices has
n(n-1)/2 edges, and is denoted by Kn
Complete Graph
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Complete graphs on n vertices, for n between 1 and 12, are shown below along with the numbers of edges:
Examples
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A subgraph of a graph G is a graph whose vertex set is a subset
of that of G, and whose adjacency relation is a subset of that of G
restricted to this subset.
Subgraph
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Graph coloring is a special case of graph labeling; it is an
assignment of labels traditionally called "colors" to elements
of a graph subject to certain constraints.
Graph coloring
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In its simplest form, it is a way of
coloring the vertices of a graph such
that no two adjacent vertices share the
same color; this is called a vertex
coloring. Similarly, an edge coloring
assigns a color to each edge so that
no two adjacent edges share the same
color, and a face coloring of a planar
graph assigns a color to each face or
region so that no two faces that share
a boundary have the same color.
Graph coloring
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Example: 2-coloring of the edges of a complete graph
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The Ramsey number is the smallest integer n such that in
any 2-coloring of the edges of a complete graph on n vertices
by red and blue, there either is a red (i.e., a complete subgraph
on k verticies, all of whose edges are colored red), or there is a
blue .
Ramsey(1930) showed that is finite for any two integers k and l.
The Ramsey Number ( , )R k l
( , )R k l
nK
lK
kK
( , )R k l
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If , then . Thus for all .
A lower bound for the diagonal Ramsey numbers ( , )R k k
12
2 1
kn
k
/ 2( , ) 2kR k k 3k ( , )R k k n
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Proof:
Consider a random 2-coloring of the edges of ;
For any fixed set R of k vertices ;
Let be the event : the induced subgraph of on R is monochromatic
Clearly,
There are possible choices for R
P(at least one of the events occurs) )
P(no event occurs) > 0
There is a 2-coloring of the edges of without a monochromatic
A lower bound for the diagonal Ramsey numbers ( , )R k k
nK
RA nK
n
k
RA
RA
nK
kK
( , )R k k n
12
( ) 2
k
RA
12
2 1
kn
k
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Proof:
Note that and we take , then
and hence for all .
This simple example demonstrates the essence of the probabilistic method.
To prove the existence of a good coloring we do not present one explicitly,
but rather show, in a nonconstructive way, that it exists.
A lower bound for the diagonal Ramsey numbers ( , )R k k
3k / 22kn
2
1 / 212
/ 2
22 1,
! 2
kk k
k
n n
k k
/ 2( , ) 2kR k k 3k
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“Erdős asks us to imagine an alien force, vastly more powerful than us,
landing on Earth and demanding the value of R(5, 5) or they will destroy
our planet. In that case, he claims, we should marshal all our computers
and all our mathematicians and attempt to find the value. But suppose,
instead, that they ask for R(6, 6). In that case, he believes, we should
attempt to destroy the aliens. ”
—Joel Spencer
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Contents
The Basic Method
The Probabilistic Method
The Ramsey Number
Linearity of Expectation
Basics
Splitting Graphs
The Probabilistic Lens:
High Girth and High Chromatic Number
( , )R k l
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Let be random variables, .
Linearity of expectation states that :
The power of this principle comes from there being no restrictions on the
dependence or independence of the .
Basic
1,..., nX X1 1 n nX c X c X
1 1 n nE X c E X c E X
iX
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In algebra and particularly in group theory, a permutation of a set S is
defined as a bijection from S to itself (i.e., a map S → S for which every
element of S occurs exactly once as image value). To such a map f is
associated the rearrangement of S in which each element s takes the
place of its image f(s).
Thus there are six permutations of the set {1,2,3}, namely [1,2,3], [1,3,2],
[2,1,3], [2,3,1], [3,1,2], and [3,2,1].
Permutation
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Let be a random permutation on {1,…,n}, uniformly chosen.
Let be the number of fixed points of .
To find , we decompose , where is the
indicator random variable of the event . Then
so that
Example
1 nX X X iX
( )i i
X
EX
1 11EX
n n
1
P ( )iEX i in
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In application we often use the fact that there is an event
in the probability space for which and an event
for which .
X EX
X EX
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A tournament is a directed
graph (digraph) obtained by
assigning a direction for each
edge in an undirected complete
graph. That is, it is a directed
graph in which every pair of
vertices is connected by a
single directed edge.
Tournament
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A Hamiltonian path is a path in a graph which visits each vertex exactly
once.
A Hamiltonian path (black) over a graph (blue).
Hamiltonian Paths
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Theorem: There is a tournament T with n players and at least
Hamiltonian paths.
Proof.
In the random tournament, let X be the number of Hamiltonian paths.
For each permutation , let be the indicator random variable for
giving a Hamiltonian path.
—that is, satisfying
Then and
Thus some tournament has at least Hamiltonian paths.
Hamiltonian Path in Tournament T
( 1)!2 nn
X
( ( ), ( 1)) for 1 .i i T i n
X X( 1)!2 .nEX EX n
EX
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Contents
The Basic Method
The Probabilistic Method
The Ramsey Number
Linearity of Expectation
Basics
Splitting Graphs
The Probabilistic Lens:
High Girth and High Chromatic Number
( , )R k l
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In graph theory, an independent set is a set of vertices in a graph, no
two of which are adjacent. That is, it is a set I of vertices such that for
every two vertices in I, there is no edge connecting the two. Equivalently,
each edge in the graph has at most one endpoint in I. The size of an
independent set is the number of vertices it contains.
Independent set
A maximum independent
set is a largest independent
set for a given graph G and
its size is denoted α(G).
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A bipartite graph (or bigraph) is
a graph whose vertices can be
divided into two disjoint sets U
and V such that every edge
connects a vertex in U to one in
V; that is, U and V are
independent sets.
Bipartite Graph
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Theorem : Let be a graph with n vertices and e edges. Then G
contains a bipartite subgraph with at least e/2 edges.
Proof.
Let be a random subset such that ;
Set ;
Call an edge {x,y} crossing if exactly one of x,y are in T ;
Let X be the number of crossing edges ;
We decompose
Where is the indicater random variable for {x,y} being crossing .
Theorem
( , )G V E
T V
B V T
{ , }
,xy
x y E
X X
xyX
P 1/ 2x T
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Theorem : Let be a graph with n vertices and e edges. Then G
contains a bipartite subgraph with at least e/2 edges.
Proof.
Then
Thus for some choice of T, and the set of those crossing
edges form a bipartite graph.
Theorem
( , )G V E
/ 2X e
1
2xyEX
{ , } 2xy
x y E
eEX EX
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Contents
The Basic Method
The Probabilistic Method
The Ramsey Number
Linearity of Expectation
Basics
Splitting Graphs
The Probabilistic Lens:
High Girth and High Chromatic Number
( , )R k l
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In graph theory, the girth of a graph is the length of a shortest cycle
contained in the graph. If the graph does not contain any cycles, its girth
is defined to be infinity. For example, a 4-cycle (square) has girth 4. A
grid has girth 4 as well, and a triangular mesh has girth 3. A graph with
girth >3 is triangle-free.
Girth
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The chromatic number of a graph G is the smallest number of colors
needed to color the vertices of G so that no two adjacent vertices share
the same color, i.e., the smallest value of k possible to obtain a k-coloring.
Chromatic Number
( )G
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Theorem : For all k,l there exists a graph G with girth(G) > l and .
Proof.
Fix ;
Let with (That is, G is a random graph on n
vertices chosen by picking each pair of vertices as an edge randomly
and independently with probability p) ;
Let X be the number of circuits of size at most l. Then
as .
Theorem (Erdős [1959])
( )G k
1/ l
~ ( , )G G n p1p n
1l
3 3
!( )
( )! 2 2
il li
i i
n nEX p o n
n i i i
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For all k,l there exists a graph G with girth(G) > l and .
Proof.
In particular,
Set so that
Let n be sufficiently large so that both these events have probability
less than 1/2.
Theorem (Erdős [1959])
( )G k
(3/ ) lnx p n
P / 2 (1)X n o
2 ( 1) / 2P ( ) (1 ) (1).
xx
p xn
G x p ne ox
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For all k,l there exists a graph G with girth(G) > l and .
Proof.
Then there is a specific G with less than n/2 cycles of length less
than l and with
Remove from G a vertex from each cycle of length at most l.
This gives a graph with at least n/2 vertices. has girth
greater than l and Thus
To complete the proof, let n be sufficiently large so that this is
greater than k.
Theorem (Erdős [1959])
( )G k
1( ) 3 ln .G n n
*G *G
*( ) ( ).G G *
*
* 1
/ 2( ) .
( ) 3 ln 6ln
G n nG
G n n n
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Vielen Dank für Ihre Aufmerksamkeit!