This preview shows page 1. Sign up to view the full content.
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
JC3,4'
(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.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory, Angles

Click to edit the document details