Lecture Notes #11 for Ma/CS 6a October 28 and 30, 2013
Consequences of Knigs Theorem
o
Corollary 1 of Knigs Theoem. A bipartite graph G that is regular of degree d > 0
o
has a perfect matching.
Proof : Let X and Y be subsets of V (G) so that each edge of
Lecture Notes #13 for Ma/CS 6a November 6, 2013
Remarks on Maxow-Mincut
Prosition 1. An unaugmentable feasible ow f (i.e. one for which there are no
f -augmenting paths from s to t) is a maxow (a maximum strength feasible ow).
Proof : Review the proof of
Lecture Notes for Ma/CS 6a October 16, 2013
Some easy and some hard problems about graphs
Suppose G is a large graph. For example, you may imagine that G has 10000 or so
vertices and average degree 100 or 200. Here are four questions I might ask about G.
Lecture Notes #12 for Ma/CS 6a November 4, 2013
Augmenting paths and the Maxow-Mincut Theorem
Theorem 1. (Maxow-Mincut) Given a network with digraph D, source s, sink t, and
capacity function c there exists a ow f0 from s to t and a cut (X0 , Y0 ) separat
Lecture Notes for Ma/CS 6a October 30, 2013
The integers
It is necessary to know the axioms (rules or laws of algebra) for working with the
integers . . . , 3, 2, 1, 0, 1, 2, 3, . . . Axioms for the natural numbers 1, 2, 3 . . . are given
in the text (Big
Lecture Notes #10 for Ma/CS 6a October 25, 2013
Matchings in bipartite graphs, continued
A matching is a graph M all of whose vertices have degree 1. By a matching in a graph
G we mean a subgraph M that is a matching. A matching M is a maximum matching
in
Lecture Notes #9 for Ma/CS 6a October 21 & 23, 2013
Notation: We will start writing V (G) and E (G) for the vertex and edge sets of a
graph G.
Cheapest (or minimal) spanning trees
Suppose each edge of a graph has a nonnegative cost. That is, we are given
Lecture Notes #8 for Ma/CS 6a October 18, 2013
The greedy algorithm, cont.
Theorem 1. Let k be a positive integer. If G is a nite graph with all vertices of degree
k and at least one vertex of degree < k , then (G) k .
Proof : We use Problem 2(iv) of Set
Lecture Notes for Ma/CS 6a October 9, 2013
Binomial numbers
We use the term n-set to mean a set of n elements. Sometimes we say a set of
cardinality n, or of size n, to mean the same thing. Also, we write |X | = n to mean that
X is an n-set (or of cardina
Lecture Notes for Ma/CS 6a October 14, 2013
Introduction to graphs
Graph theory has many technical terms. Unfortunately, perhaps, terminology diers
from one author or school of graph theory to another. For example, Biggs says a graph
has vertices and edge
Lecture Notes for Ma/CS 6a October 7, 2013
Primality testing
For the RSA cryptosystem, we need a large (perhaps 300-digit) integer m and the
value of (m). We propose to take m = pq where p and q are distinct primes of about 150
digits each.
How to we nd s
Lecture Notes for Ma/CS 6a October 4, 2013
Fermats little theorem and the Euler phi-function
Every one knows that the square of an odd number is odd and the square of an even
number is even. This is the case p = 2 of the following theorem.
Theorem 1. (Fer
Lecture Notes for Ma/CS 6a September 2, 2013 Revised Sept. 3
Congruences
For integers a, b, m, the statement a b (mod m) (read a is congruent to b modulo
m) means m | (b a).
It may seem silly to introduce notation that means the same thing as m divides b
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