HW #2: due Oct 9

1. Let n be an integer. Prove that n³-n is a multiple of 6.

2. Let G be a connected graph and v a vertex of degree 1. Using the vertex-based definition we came up for of connectedness, prove that the deletion G\v is also connected.
("Let w,x be any two vertices of G\v. We want to show that there exists a chain in G\v...")

3. Let G be a graph and v a vertex of degree 1. Assume that the deletion G\v has a nontrivial decomposition. Construct a decomposition of G.
("Let V₁, V₂ be a nontrivial decomposition of G\v. We will use this to build one of G itself...")

4. What's the relation between #2 and #3?

From [FP]:

1.4 #56,58,59,67
1.5 #84,90
1.6 #93,99,100,102