184
PERSI DIACONIS
Suppose, as is the case for the examples in this paper, that the Markov chain
is reversible: (x)K(x, y) = (y)K(y, x). Let L2 () be cfw_g : X R with inner
product
g, h =
g(x)h(x)(x).
x
Then K operates on L2 by
Kg(x) =
g(y)K(x, y).
Revers
MATH 350: Graph Theory and Combinatorics. Fall 2013.
Assignment #3: Network ows, covers and Ramseys theorem.
Let G be a directed graph and for each edge e let (e) 0 be an
integer, so that for every vertex v,
(e) =
(e)
1.
e (v)
e + (v)
Show there is a list
3.
Show that R(3, 4) = 9.
Solution: We start by showing that R(3, 4) 9. Suppose for a contradiction that G is a graph on 9 vertices containing no independent set of size
3 and no clique of size 4. By Theorem 11.1 R(3, 3) R(3, 2) + R(2, 3) = 6.
If deg(v) 6
194
PERSI DIACONIS
Alas, one is not always so lucky. The Metropolis chain of (5.1) has the form
P f (x) = m(x)f (x) + h(x, y)f (y)dy. The multiplier m(x) leads to a continuous
spectrum. One of our discoveries [24, 25, 59] is that for many chains, this can
u
u u f u | Cs qx Cs | Cs 0
u v w r o tge w
Rbhlmhfnqphu |fhVRygmw w
u xfh0yu~wuhxbnheqwhhnnhqennI0u v Tyowy~nhhqunspohAVRxygw
m ~ me ~ re vutg r e g t
we r w
ymw qu Tyowy~nhhqunfspohrlmnfzRhf|wzsw$qznspAhp hVbdq!dhwu
m
we r w rku m vu oku m o u
THE MARKOV CHAIN MONTE CARLO REVOLUTION
199
This is a useful, explicit bound but it is often o, giving the wrong rate of
convergence by factors of n or more in problems on the symmetric group Sn . A
host of more complex techniques can give better results.
THE MARKOV CHAIN MONTE CARLO REVOLUTION
189
are chosen from the uniform distribution and f : X (n, ) R is a function, we may
approximate
(4.1)
f (x)dx by
X (n, )
1
k
k
f (Xi ).
i=1
Motivation for this task and some functions f of interest will be given at
v
u
w
Figure 1: Counterexample for Problem 1a).
MATH 350: Graph Theory and Combinatorics. Fall 2013.
Assignment #1: Paths, Cycles and Trees. Solutions.
1.
For each of the following statements decide if it is true or false, and
either prove it or give a co
MATH 350: Graph Theory and Combinatorics. Fall 2013.
Assignment #2: Bipartite graphs, matching and connectivity.
1.
Show that every loopless graph G contains a bipartite subgraph with
at least |E (G)|/2 edges.
Solution: Let a partition (A, B ) of V (G) be
MATH 350: Graph Theory and Combinatorics. Fall 2013.
Assignment #5: Planar graphs. Solutions.
1.
A graph G is outerplanar if it can be drawn in the plane so that
every vertex is incident with the innite region. Show that a graph G is
outerplanar if and on
MATH 350: Graph Theory and Combinatorics. Fall 2013.
Assignment #4: Matchings and coloring. Solutions.
1. Let G be a graph and Z V (G). Show that the following are equivalent:
(i) G has a matching covering Z , and
(ii) for every X V (G) there are at most