Assignment 6
Prof. John MacDonald
Spencer Frei
March 12, 2014
1. 4.5.11. Count. For each of the regular solids, take the number of vertices V , subtract the number
of edges E, and add the number of faces F . For each regular solid, what do you get? See the
scanned image at the bottom.
2. 4.5.16. Cube slices. Consider slicing the cube with a plane. What are all the different-shaped
slices we can get? One slice, for example, could be rectangular. What other shaped slices can we
get? Sketch both the shape of the slide and show how it is a slice of the cube. See the scanned
image at the bottom.
3. 6.1.9. Scenic drive. It is not possible to obtain a trip that traverses every road in the park
starting from the entrance because there are vertices in the park with odd degree. Mr. Big Tree
and the Entrance both have odd degree and thus there is no chance of being able to visit every
edge exactly once.
4. 6.1.14. Walking the dogs. See the scanned image.
5. 6.1.16. Snow Job. See the scanned image.
6. 6.1.24. Path to proof. There are two vertices with odd degree and the rest have even degree.
So, if we add an edge between the two vertices with odd degree, the degree of each will increase
by one, and since an odd number plus one is an even number, this will make all degrees in the
graph now have even degree. Thus there exists an Eulerian circuit in the newly formed graph.
7. 6.1.26. Degree day. For the first one, there are 3 vertices with degree 1, 3 vertices with degree
3, 3 vertices with degree 2, and 1 vertex with degree 4. So, the sum of degrees is 22, and the
number of edges is 11. For the second, the sum of degrees is 18, and the number of edges is
9. For the third, the sum of degrees is 24, and the number of edges 12. We see each time that
the sum of degrees of a graph is double the number of edges: this is because when we sum the
degrees of a graph, we count each edge twice; an edge is a pair of vertices (u, v) and since u is
connected to v and v is also connected to u, we will count two degrees for each edge.
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