AMS 301.3
Fall, 2009
Homework Set # 2 — Solution Notes
# 4, Supplement II:
A graph with
n
vertices can have at most
(
n
2
)
=
n
(
n

1)
2
vertices; this would be the case
of a complete graph,
K
n
, in which all
n
vertices have degree (
n

1). Thus, if we have 3 vertices, there are
at most 3 edges, if we have 4 vertices, there are at most 6 edges, etc. If we have
n
= 10, there are at most
(
10
2
)
= 45 edges, so it is impossible to have a 10vertex graph with 50 edges. (NOTE: We are assuming here
that it is a simple graph, not a multigraph; in a multigraph, even if there is only ONE vertex, it is possible
to have 50 (or 50,000) edges — just make them all selfloops!) If we have
n
= 11, there are at most
(
11
2
)
= 55
edges (and
K
11
has exactly 55 edges). Thus, with 11 vertices it is possible to have 50 edges (e.g., take
K
11
and delete any 5 edges), and it is not possible to have 50 edges if there are fewer than 11 vertices.
#6, 1.3: The number of edges in
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 Spring '08
 ARKIN
 Graph Theory, Planar graph, edges, complete graph

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