Chapter 9
261
5.
(a)
JC4,
the complete graph on 4 vertices has this degree sequence.
(b) No such graph exists because the sum
of
the degrees in a graph must be even, but the sum here is
15, which is not even.
(c) Draw two triangles, with vertices
AI,
Bl,
C
l
and
A2,
B
2,
C
2,
respectively. Then add edges
AIA2,
B
I
B
2,
C
I
C
2 .
This gives a graph with the given degree sequence.
(d) Drawing a graph by the approach
of
part (c), but starting with two squares, gives a graph with the
given degree sequence.
6.
(a) The sum
of
the degrees is 41, an odd number, contradicting Proposition 9.2.5.
(b) Suppose such a graph existed.
It
would have nine vertices two
of
which have degree
8.
Hence,
these two would be joined to all vertices except themselves, and thus every vertex would have
degree at least
2.
This contradicts the fact that there is a vertex
of
degree
1.
7.
(a) This is the complete bipartite graph
JC
3
,4'
(b) This is not bipartite.
It
contains an odd cycle (the vertices along the top and right side).
(c) This is bipartite, but not complete. There are black and white vertices not joined by an edge.
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 Summer '10
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 Graph Theory, Angles, Vertex, Bipartite graph, Complete bipartite graph

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