This assignment explores two fundamental concepts in mathematics: graph coloring and the pigeonhole principle. It demonstrates how a unique path exists between any two vertices in a tree, proving its acyclicity. The assignment further illustrates the two-colorability of all trees by applying induction and considering a leaf vertex and its adjacent vertex. Finally, it uses the pigeonhole principle to show that within a set of four integers, there must be at least two with the same remainder when divided by 4.