Math 2200 Homework 9
Due Date: November 27
Please write neatly and leave enough space so I can write comments. Write the theorems, and, as always,
use complete sentences when writing proofs. When I ask for examples of graphs with various properties,
please draw pictures (instead of writing down two sets and a function) and explain why your example works!
If you get stuck, come to office hours, make an appointment, or e-mail me.
Degree and Simple Graphs
Let G be a graph. Recall that the degree deg v of a vertex v is the number of edges that have v as an
endpoint, with loops counted twice.
Exercise 1. Prove the following theorem:
Theorem. Let G = (V, E, φ) be a graph. Then the sum of the degrees of all vertices in G is equal to twice
the number of edges of G, namely
!
deg(v) = 2 · |E|.
v∈V
A graph is said to have multiple edges if there are two distinct edges connecting the same two vertices,
as in the graph •
• . A graph is said to be simple if it has no loops and no multiple edges.
Exercise 2. Prove the following theorem:
Theorem. Let G = (V, E, φ) be a simple graph and let n = |V | be the number of vertices of G. Then |E|,
the number of edges of G, satisfies the inequality
" #
n
|E| ≤
.
2
Give examples of simple graphs with |V | = 3, 4, 5 where the inequality is in fact an equality, namely
$ %
|E| = |V2 | . (Simple graphs where the inequality is an equality are called complete graphs.) Also, give an
example of a non-simple graph with four vertices where the inequality fails.
Connectivity
Exercise 3. Prove the following theorem:
Theorem. Let H1 , H2 be connected subgraphs of a graph G and suppose H1 and H2 have a common vertex.
Then H1 ∪ H2 is connected.
Also, give an example to show this may be false if H1 is not connected, and give an example to show this
may be false if H1 and H2 do not have a common vertex.
Exercise 4. Prove the following theorem:
Theorem. Let H1 , H2 be connected subgraphs of a graph G. Then H1 ∩ H2 is connected.
1