Section 50
1.
We noticed that a graph with more than two vertices of odd degree
cannot have an Eulerian trial, but connected graphs with zero or two
vertices of odd degree do have Eulerian trails.
The missing case is
connected graphs with exactly one vertex of odd degree. What can you
say about those graphs
?
There aren°t any graphs with exactly one vertex of odd degree. The
sum of the degrees of the vertices is equal to twice the number of edges.
If there were exactly one vertex of odd degree, the sum of the degrees
of the vertices would be an odd number.
2.
A domino is a
2
°
1
rectangular piece of wood.
On each half of the
domino is a number, denoted by dots. In the °gure we show all
°
5
2
±
=
10
dominoes we can make where the numbers on the dominoes are all
pairs of values chosend from
f
1
;
2
;
3
;
4
;
5
g
(
we do not include dominoes
where the two numbers are the same
).
Notice that we have arranged
the ten dominoes in a ring such that, where two dominoes meet, they
show the same number.
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 Spring '08
 STAFF
 Graph Theory, Vertex, Eulerian, odd degree

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