Euler's Theorems

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Euler's Theorems
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Our problem
The road crew wants to clear the streets in the
Holbrook Estates subdivision.
Can they do this by going down each street only
once?
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Our approach

• We represent the important information in the
  problem (streets and corners) by a graph . . .

. . . and then discard our original description.
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Our question

• Now that we have the graph, can we find an Euler
  circuit (i.e. a circuit that that travels across
  every edge once and only once)?

start here
We could try and try to see if we can find one,
or we can use the power of graph theory.
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Our solution

• Our answer lies in noting what happens when
  traversing a circuit.
• Since we must pass over every edge, we must go
  through every vertex at least once.
• Each time we visit a vertex, we come in on one
  edge and leave on another.

This leads us to a theorem.
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Eulers First Theorem

• The statement
• (a) If a graph has any vertices of odd degree,
  then it cannot have an Euler circuit.
• (b) If a graph is connected and every vertex has
  even degree, then it has at least one Euler
  circuit.
• Using the theorem
• We need to check the degree of the vertices.
• Note that this does not help us find an Euler
  circuit, it only tells us if there is one.

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Applying Eulers First Theorem

• We start with our graph and check the degrees of
  the vertices. (Recall the degree of a vertex is
  the number of edges that connect to that vertex).
  
• If we find even one vertex of odd degree, then we
  cannot have an Euler circuit.

vertex of degree 3
So finding an Euler circuit in this graph is
impossible.
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Finding the best possible result

• So our ideal solution is impossible.
• Whats the best we can do?
• First we need some more vocabulary.
• A multigraph is a graph in which pairs of
  vertices may be connected by more than one edge
  and which may contain loops.

Ex.
In this multigraph there are two edges connecting
A and D as well as B and C. Vertex C has degree
5.
4
3
5
2
1
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The process

• Since we cant find an Euler circuit in the
  graph, we will have to travel some streets or
  sections more than once.
• To indicate this we will duplicate edges to
  indicate those sections we will retravel.
• When we are finished, we should be able to find
  an Euler circuit in the resulting graph (i.e.
  every vertex will have even degree).
• So our plan will be to eliminate all the vertices
  of odd degree.

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Holbrook Estates Solved
First we should duplicate the edge at the
entrance to indicate we will be going back out
that road.
Now find the vertices of odd degree.
Then duplicate edges until they all have even
degree.
This process is called Eulerizing the graph. We
can only duplicate existing edges, never add an
edge where one did not already exist.