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.

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.

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.

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.

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

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