**Regular Graphs and Euler Relations**

prepared by:

Joseph Malkevitch

Mathematics Department

York College (CUNY)

Jamaica, New York 11451

email: malkevitch@york.cuny.edu

1. Derive an Euler relation for 4-valent plane graphs.

2. For the face vectors below, try if possible to construct a plane 3-valent 3-connected graph with the given numbers of faces.

a.

b.

c.

3. Draw diagrams of all the graphs of the 5 Platonic solids.

4. If one erects a combinatorial square pyramid on each face of the plane graph of a cube, write down the vertex and face vectors of the resulting graph.

Hint: (The diagram below indicates what has to be done on each face.)