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Homework 9 Solutions
1. Show that there are inﬁnitely many plane graphs
G
with the following properties:
(i)
Every face of
G
is a triangle
(ii)
Every vertex is incident with 5 or 6 edges.
(Hint: An Icosahedron is one such graph, modify it by inserting some new vertices and edges
to ﬁnd more)
Figure 1: Icosahedron
Solution:
We start with the icosahedron and apply the following operation. Take each edge
and add a new vertex in the middle of it, thus splitting the original edge into two. Then in
each original face add a triangle on the three new vertices. This preserves the property that
every face is a triangle, and each new vertex becomes incident with exactly 6 edges, so both
properties are still satisﬁed. By repeating this operation, we can create inﬁnitely many such
graphs.
2. Let
G
be a connected plane graph with
v
vertices and
e
edges so that every vertex lies on
one square face, one hexagonal face, and one octagonal face. Let
f
4
,f
6
,f
8
be the number of
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 Spring '11
 RWK

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