1. Show that the join of a 3-cycle and a 5-cycle contains no but that every 2-edge coloring yields a monochromatic triangle.

2. Let and be k-critical graphs with exactly one vertex v in common, and let and be edges of and . Show that the graph U is k-critical.

3. Show that a connected -critical graph has no cut vertices. is the independence number and a graph is

-critical if (G - e) > (G) for all e in the edge set of G.

Sorry I forgot that little bit of extra instruction before.