and append to it each of the closed paths to ensure that each edge is used exactly once. Exercise 1.5 Show that for every graph G , there exists a graph H so that G is a minor of H , and for all v ∈ V ( H ), deg( v ) ≤ 3. Proof. Let G = ( V, E ) be a graph such that deg( v ) > 3 for every v ∈ V , and let w 1 ∈ V be the vertex of largest degree. Contract G by w 1 to get G 0 = ( V 0 , E 0 ). If the degree of every vertex in G 0 is still greater than 3, choose the largest vertex w 2 ∈ V 0 and contract G 0 by w 2 to get G 00 = ( V 00 , E 00 ). Proceed in this manner until the degree of every vertex is at most 3. This is clearly possible, because one can simply go contracting off vertices until there insufficiently many of them for any to have degree greater than 3. Exercise 1.11 Recall that b 0 ( G ) is the number of connected components of a graph G . Show that b 0 ( G ) = dim V - dim Im( d ) Lemma Let G = ( V, E ) be a graph. If G has no cycles, then G has | V | - | E | connected components. Proof. Since G has no cycles, G consists of (possibly disconnected) trees, meaning that each connected component G i = ( V i , E i ) of G satisfies | E i | = | V i | - 1, or 1 = | V i | - | E i | . Summing over all of the connected components gives k = | V | - | E | , so G has k connected components.
- Spring '08