eulers and hamilton theory

8/6/2019 eulers and hamilton theory

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Euler Paths and Circuits

DefinitionDEF: AnEuler path in a graph G is a simple

path containing every edge in G. AnEuler

circuit(orEuler cycle) is a cycle which is

an Euler path.

NOTE: The definition applies both to

undirected as well as directed graphs of alltypes.


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Hamilton Paths and Circuits

DefinitionDEF: A Hamilton path in a graph G is a pathwhich visits ever vertex in G exactly once. A

Hamilton circuit(orHamilton cycle) is a cycle

which visits every vertex exactly once, exceptforthe firstvertex, which is also visited at the

end of the cycle.

NOTE: Again, the definition applies both toundirected as well as directed graphs of all

types.


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An Euler Circuit is a cycle of an undirected graph, that traverses every

edge of the graph exactly once, and ends at the same node from

which it began.

Euler's Theorem: A connected graph G possesses an Euler circuit if and

only ifG does not contain any nodes of odd degree.

Proof of Euler's theorem: Assume that G has zero nodes of odd

degree. It can then be shown that this is a necessary and a sufficient

condition for an Euler circuit to exist.

Part 1: It is necessary because any Euler circuit drawn on the graph

must always enter a node through some edge and leave through

another and all edges on the graph must be used exactly once.

Thus, an even number of incident edges is required for every

node on the graph.


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1. Pick any vertex to start.

2. From that vertex pick an edge to traverse, considering following rule:

never cross a bridge of the reduced graph unless there is no other choice.

3. Darken that edge, as a reminder that you can't traverse it again.

4. Travel that edge, coming to the next vertex.

5. Repeat 2-4 until all edges have been traversed, and you are back at the

starting vertex.

By reduced graph we mean the original graph minus the darkened (already

used) edges.

A bridge of a graph G is an edge whose deletion increases the number of

components ofG.

Fleury's Algorithm: O(E)?


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Fleury's Algorithm in Action

Pick any vertex (e.g. F)

Take F to C (arbitrary)Take C to D (arbitrary) Take D to A (arbitrary)

Take A to C.

Can't go to B: that edge is a

bridge of the reduced graph,

and there are two other choices.

How can we recognize a bridge efficiently?

In the original graph, AB was not a bridge.

Can we preprocess the graph in O(E) time

identifying bridges and building a structure that

can be updated in constant time with each

reduction?

A bridge is not a local

property (i.e. if edge EF

existed then AB would not

be a bridge).


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Dodecahedral GraphIs it Hamiltonian? If so, find the Hamiltonian CycleA

B

CD

E NO

F

G

H

P

JK

M

Z

V

W

X

Y

L


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Graph ColoringConsider a fictional continent.


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Map ColoringSuppose removed all borders but still wantedto see all the countries. 1 color insufficient.


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Map ColoringSo add another color. Try to fill in everycountry with one of the two colors.


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Map ColoringPROBLEM: Two adjacent countries forced tohave same color. Border unseen.


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Map ColoringSo add another color:


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Map ColoringInsufficient. Need 4 colors because of thiscountry.


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Map ColoringWith 4 colors, could do it.

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eulers and hamilton theory