# hw9sol - Homework 9 Solutions 1 Show that there are...

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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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hw9sol - Homework 9 Solutions 1 Show that there are...

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