1. What is the next line of the proof?
a). Assume the result holds for graphs with k vertices.
b). Assume the result holds for graphs with k edges.
c). Assume the result holds for graphs with k+1 vertices.
d). Assume the result holds for graphs with k+1 edges.
e). If p =1, note that the theorem holds.
f). If q = 1, note that the theorem holds.

2. What is the next line of the proof?
a). Assume the result holds for graphs with k vertices.
b). Assume the result holds for graphs with k edges.
c). Assume the result holds for graphs with k+1 vertices.
d). Assume the result holds for graphs with k+1 edges.
e). If p =1, note that the theorem holds.
f). If q = 1, note that the theorem holds.

3. What is the next line of the proof?
a). Let G be a planar graph with k + 1 vertices.
b). Let G be a planar graph with k vertices and add a vertex v to it.
c). Let G be a planar graph with k – 1 vertices.
d). Let G be a planar graph with k – 1 vertices and add a vertex v to it.
e). Assume a planar graph G can be colored in 5 colors.
f). Assume a planar graph G can be colored in 6 colors.

4. What is the next line of the proof?
a). Assume G can be colored in 5 colors.
b). Assume G can be colored in 6 colors.
c). Add a vertex v to G and consider G + v.
d). Delete a vertex v from G and consider G – v.
e). Add an edge e to G and consider G + e.
f). Delete an edge e from G and consider G – e.

5. Final Project Ideas: on colorings
Coloring algorithms
Results on coloring number
Clique number and perfect graphs
Edge colorings
Probably lots more……

6. More Final Project Ideas:
Line Graphs
Matchings and assignment problems
Chess and graph theory
“Pebbling”
Etc, etc, etc…