Introduction to Hamilton Circuits
The photographer needs to start and
end in the same place.
Number of edges that
need to be crossed?
Total cost of the trip
Need a Euler Circuit with
no odd vertices.
11x25 = 275 to cross
each bridge once
Had 4 odd vertices. I
added another link
between L and D and
another link between R
and B.
Plus another 2x25 =
50 for the two
additional links
representing retrace)
Total: $325
The photographer has the freedom to
choose any starting and ending point
he choses.
Need a Euler path with
exactly 2 odd vertices.
11x23 =275
1 x 25 = 25
Had 4 odd vertices, so I
added a link between R
and B, converting those 2
odd vertices to even
vertices.
Total: $300
The photographer must start his trip a
point B and end the trip at point L.
Need a Euler path where
B and L remain odd and
are the only odd vertices.
11x 25 = 275
3x 23 = 75
I added links between C
and D, between R and A,
to make R and D even.
These changes made A
and C odd, so make
those vertices even
again, I added a third link
connecting A and C.
Total: $325
Hamilton Paths and Circuits
A Hamilton Path
Is a path that visits each vertex of the graph once and only once.
A Hamilton Circuit
Is a path that visits each vertex of the graph once and only once and ends at the
starting point.
Look at the following graphs:
How are Euler paths and Hamilton paths related?
a
Euler
Circuit
Euler Path
Hamilton
Circuit
Hamilton
Path
Yes
No
Yes
Yes
b
c
d
e
f
Conclusion:
The existence of an Euler path or Circuit tells us nothing about the
existence of a Hamilton path or circuit.
Dirac’s theorem:
If a connected graph has N vertices (N>2) and all of them have degree bigger or
equal to N/2, then the graph has a Hamilton Circuit.
A complete graph
is a graph with N vertices in which every pair of distinct vertices is joined by an
edge.
Examples of complete graphs
Can you tell how many Hamilton Circuits on in each figure above?
Hamilton Path Puzzles