Discrete 121206
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Transcript of Discrete 121206
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Discrete Math
Week 15 December 6, 2012
Shimizu
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Theorems
The Handshake Theorem: If G is any graph, thenthe sum of the degrees of all of the vertices of G
equals twice the number of edges of G.
In other words, If the vertices of G are (integer 1, 2, 3, ), then
deg 2(# )= .
Corollary to the Handshake Thm: The total degreeof a graph is even
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Practice
Is it possible to draw a graph with the specifiedproperties? If so, draw an example. If not, explain
why.
4 vertices of degrees 1, 1, 2, and 3. 4 vertices of degrees 1, 1, 3, and 3.
Simple graph, 4 vertices of degree 1, 1, 3, and 3.
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Proposition 10.1.3. In any graph there are an evennumber of vertices of odd degree.
Proof:
(start of ) Suppose G is any graph and G has n vertices of odd
degree and m vertices of even degree (n and m are
positive integers).
What do we have to show?
What else do we know about G?
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Definitions
Let G be a graph, and let v and w be vertices in G.
A walk from v to w is a finite alternating sequence ofadjacent vertices and edges of G.
A walk is of the form0 .
The vs represent vertices, the es representedges, 0 , , and the endpoints of
are and (for all 1,2,3, , )
A trivial walk from v to v consists of the single vertexv.
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Definitions
A trail from v to w : a walk that does not contain arepeated edge.
A path from v to w: a trail that does not contain a
repeated vertex.
A closed walk: a walk that starts and ends at thesame vertex.
A circuit: a closed walk that contains at least one
edge and does not contain a repeated edge.
A simple circuit: a circuit that does not have any
other repeated vertex except the first and last vertex.
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More definitions
Let G be a graph. A graph H is a connectedcomponent of a graph G iff
1. H is a subgraph of G;
2. H is connected; and
3. No connected subgraph of G has H as a subgraph andcontains vertices or edges that are not in H.
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More definitions
Let G be a graph. Two vertices v and w areconnected iff there is a walk from v to w.
The graph G is connected iff given any two vertices
v and w in G, there is a walk from v to w.
Practice: Is this graph connected?
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7 Bridges of Knigsberg
Is it possible for a person to take a walk around thetown of Knigsberg, start and end at the same
location and cross each of the 7 bridges exactly
once?
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More definitions
Let G be a graph. An Euler circuit for G is a circuitthat contains every vertex and every edge of G.
Is there an Euler circuit for the Konigsberg bridges
trail?
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7 Bridges of Knigsberg
Is it possible for a person to take a walk around thetown of Knigsberg, start and end at the same
location and cross each of the 7 bridges exactly
once?
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More definitions
Let G be a graph. A Hamiltonian circuit for G is asimple circuit that includes every vertex of G.
Is there a Hamiltonian circuit for the Konigsberg
bridges trail?
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Theorem 10.2.2
If a graph has an Euler circuit, then every vertex ofthe graph has a positive even degree.
Contrapositive of Thm 10.2.2:
If some vertex of a graph has odd degree, then the
graph does not have an Euler circuit.
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Theorem 10.2.3
If a graph G is connected and the degree of everyvertex of G is a positive even integer, then G has an
Euler circuit.
Does this graph have an Euler circuit?
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Theorem 10.2.4
A graph G has an Euler circuit iff G is connected andevery vertex of G has positive even degree.
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Homework
10.2/ (1 20)odd
Due next Wednesday, December 12.
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Color Square Game
C1 C2 C3 C4
R1
R2
R3
R4
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Chapter 10: Graphs and Trees
Definitions A graph G consists of 2 finite sets:
1. a non-empty set V(G) ofvert ices
2. a set E(G) ofedges. Each edge is associated with a set
consisting of one or two vertices, called its endpoints.
The correspondence from edges to endpoints is called the
edge-endpoint function.
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This is where we stand
Week # Mon. Wed. Chapters we are supposed to cover
according to the department syllabus
10 Class
cancelle
d
Class
cancelle
d
11 11/5 11/7 6.1 6.2 6.3
12 11/12 11/14 7.1 7.2 7.3
13
Thanks-
giving
week
11/19
mid-term
11/21 8.1 8.2 8.3
14 11/26 11/29 10.1 10.2
15 12/3 12/5 This means we have to do 1 to 2 sections
per class.
16 12/10 12/12 This means we will have homework over
Thanksgiving break.
17 12/17