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The full question is:

In each case, give the values of $r$, $e$, or $v$ assuming that the graph is planar. Then draw a connected planar graph with the property, if possible.

The values I was given were 6 vertices all of degree 4 and another problem with the given values 5 regions and 10 edges.

According to Euler's formula $r = e-v+2$, there should be 12 edges and 8 regions for the first problem and 7 vertices for the second problem. However I can only come up with a graph that has 7 regions for the first problem and 6 vertices in the second one. What am I doing wrong and how should I tackle drawing these problems?

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Xander Henderson
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Shawn Li
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    Don't forget the unbounded face. – Bob Jones Aug 31 '17 at 02:23
  • I just started this course in college but can I get a brief explanation of what the unbounded face is? – Shawn Li Aug 31 '17 at 02:25
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    @ShawnLi It is the part of the paper that is outside of the picture you drew. – Xander Henderson Aug 31 '17 at 02:26
  • Your first graph has 8 regions, your second graph has 6 regions. You're forgetting to count the big outer region. – bof Aug 31 '17 at 02:27
  • I see, so one of the regions that are counted is external of the graph. Somewhat confusing but I think I understand – Shawn Li Aug 31 '17 at 02:28
  • @ShawnLi Exactly! It might be useful to know that the formula comes from looking at faces, edges, and vertices on polyhedra. Planar graphs correspond (roughly) to things that are homeomorphic to a sphere, such as a cube. Remove one face of the cube, then flatten it out---you get a planar graph. The face that you removed is the unbounded face of the graph. – Xander Henderson Aug 31 '17 at 02:59

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