Using Kruskal's algorithm, construct a minimum spanning tree. Be sure to list the edges you choose in the order that you choose them. In the case of a tie, choose edges alphabetically.

Determine $($with proof$)$ necessary conditions on graphs $G$ and $H$ so that $GH$ is a tree. Hint: It may help your intuition to look at small graphs, say with two vertices...

Let $T$ be a tree. Prove that if $T$ has a perfect matching, then it has a unique perfect matching.

Let $G$ be a graph. Prove that the Cartesian product $G{K}_2$ has a perfect matching.