## Description

1. Show that πΎ8 can be drawn on a 2-holed torus without edges

crossing. Feel free to use the octagon model as a framework for youre

drawing:

2. Use an edge-counting argument to show that πΎ9 cannot be drawn on a

2-holed torus without edges crossing. Ingredients: For πΎ9

, you have

π = 9, π = (

9

2

) = 36. What would π have to be? What is a lower

bound on the total edge count since every region must be bounded by

at least three edges?

π

π

π

π

π

π

π

π

3. If we re-orient the arcs around the diagram from #1 so they all point

clockwise, what is the resulting value of π β π + π?

4. In terms of π β {2,3,4,5,6, β¦ }, how many tournaments are there with

the node set π = {1,2,3, β¦ , π}? This is equivalent to asking for how

many ways are there to orient the edges of πΎπ with vertex set

{1,2,3, β¦ , π}.

5. Let π β {3,4,5,6, β¦ } be fixed. Show that there are exactly two

orientations of πΆπ with vertex set π = {0,1,2, β¦ , π β 1} that are

strongly connected.

π

π

π

π

π

π

π

π