CSE 20 – Discrete Mathematics

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### CSE 20 – Discrete Mathematics

Dr. Cynthia Bailey Lee

Dr. Shachar Lovett

Today’s Topics:
• Graphs
• Some theorems on graphs
Graphs
• Model relations between pairs of objects
• Basic ingredient in many algorithms: Network routing, GPS guidance, Simulation of chemical reactions,…
Graph terminology
• The fruits are the
• Graphs
• Vertices
• Edges
• Loops
• None/other/more than one
Graph terminology
• The arrows are the
• Graphs
• Vertices
• Edges
• Loops
• None/other/more than one
Graph terminology
• Is this graph
• Undirected
• Directed
• Both
• Neither
• None/other/more than one
Graph terminology
• Which of the following is not a correct graph

A.

B.

C.

D. None/other/more than one

Our first graph theorem
• Recall: in any group of 6 people, either there are 3 that form a club, or 3 that are strangers
• Any graph with 6 vertices contains either a triangle (3 vertices all connected) or an empty triangle (3 vertices not connected)
Our second graph theorem
• Let G be an undirected graph
• Degree of a vertex – number of edges adjacent to it (e.g. touch it)
• Denote it by degree(v)
• Theorem: in any undirected graph, the sum of all the degrees is even
• Try and prove yourself first
Our second graph theorem
• Theorem: in any undirected graph, the sum of all the degrees is even
• Proof:

Consider pairs (v,e) with v a vertex and e an edge adjacent to it. Create a list of all such pairs. How many elements this list has? We calculate it in two ways

• Each vertex v has degree(v) edges adjacent to it, so this list has sum of degrees many elements
• Each edge has 2 vertices adjacent to it, so this list has twice the number of edges many elements

So, sum of degrees = twice the number of edges, hence it must be even. QED.

Eulerian graphs
• Let G be an undirected graph
• A graph is Eulerian if it can

drawn without lifting the pen

and without repeating edges

• Is this graph Eulerian?
• Yes
• No
Eulerian graphs
• Let G be an undirected graph
• A graph is Eulerian if it can

drawn without lifting the pen

and without repeating edges

• Yes
• No
Eulerian graphs
• How can we check if a graph is Eulerian?
• Check all possible paths
• Stare and guess
• Be brave and do some math
Eulerian graphs
• Degree of a vertex: number of edges adjacent to it
• Euler’s theorem: a graph is Eulerianiff the number of vertices with odd degrees is either 0 or 2 (eg all vertices or all but two have even degrees)
• Does it work for and ?
Proving Euler’s theorem
• Euler’s theorem gives a necessary and sufficient condition for a graph to be Eulerian
• All degrees are even
• Two degrees odd, rest are even
• Will prove in class that this is necessary
• Take-home challenge: prove that this is also sufficient
Proving Euler’s theorem: necessary part
• Euler’s theorem (necessary part):

If a graph G is Eulerian then all degrees are even; or two degrees are odd and rest are even

Try to prove it first yourself

Proving Euler’s theorem: necessary part
• Proof of Euler’s theorem (necessary part):

Let G be a graph with an Euler path: v1,v2,v3,….,vk where (vi,vi+1) are edges in G; vertices may appear more than once; and each edge of G is accounted for exactly once.

The degree of a vertex of G is the number of edges it has. For any internal vertex in the path (eg not v1 or vk), we count 2 edges in the path (one going in and one going out). So, any vertex which is not v1 or vk must have an even degree.

If v1vk then both have odd degrees. If v1=vk is the same vertex this is also has even degree. QED.

Another example (student self-study)

Example 2
• A number x is rational if x=a/b for integers a,b.
• E.g. 3=3/1, 1/2, -3/4, 0=0/1
• A number is irrational if it is not rational
• E.g (proved in textbook)
• Theorem: If x2 is irrational then x is irrational.
Example 2
• Theorem: If x2 is irrational then x is irrational.

Assume that

• There exists x where both x,x2 are rational
• There exists x where both x,x2 are irrational
• There exists x where x is rational and x2 irrational
• There exists x where x is irrational and x2 rational
• None/other/more than one
Example 2
• Theorem: If x2 is irrational then x is irrational.

Assume that there exists x where x is rational and x2 irrational.

Try by yourself first

Example 2
• Theorem: If x2 is irrational then x is irrational.

Assume that there exists x where x is rational and x2 irrational.

Since x is rational x=a/b where a,b are integers.

But then x2=a2/b2. Both a2,b2 are also integers and hence x2 is rational.

A contracition.

Example 3
• Theorem: is irrational

THIS ONE IS MORE TRICKY.

TRY BY YOURSELF FIRST IN GROUPS.

Example 3
• Theorem: is irrational
• Assume not. Then there exist integers a,b such that
• Squaring gives

So also is rational since

[So, to finish the proof it is sufficient to show that

is irrational. ]

Example 3
• Theorem: is irrational
• is rational … is rational.
• =c/d for positive integers c,d.

Assume that d is minimal such that c/d=

Squaring gives c2/d2=6.

So c2=6d2 must be divisible by 2.

Which means c is divisible by 2.

Which means c2 is divisible by 4.

But 6 is not divisible by 4, so d2 must be divisible by 2.

Which means d is divisible by 2.

So both c,d are divisible by 2. Which means that (c/2) and (d/2) are both integers, and (c/2) / (d/2) =

Contradiction to the minimality of d.