### Part 3 - WordPress.com

```Group 10 Project Part 3
Derrick Jasso
Rodolfo Garcia
Ivan Lopez M.
Section 10.1 #36
 36. Recall that Km,n denotes a complete
bipartite graph on (m, n) vertices.
 a. Draw K4,2
 b. Draw K1,3
 c. Draw K3,4
 d. How many vertices of Km,n have degree m? degree n?
 e. What is the total degree of Km,n?
 f. Find a formula in terms of m and n for the number of
edges of Km,n. Explain.
Solution
a) K4,2
b)K1,3
Solution
c)K3,4
d) If n≠m, the vertices of Km,n are divided into two groups: one of size m and the
other of size n. Every vertex in the group of size m has degree n because each is
connected to every vertex in the group of size n. So Km,n has n vertices of degree
m. Similarly, every vertex in the group of size n has degree m since each is
connected to every vertex in the group of size m. So Km,n has n vertices of degree
m.
Solution
e) The total degree of Km,n is 2mn because Km,n has m vertices of degree n
and n vertices of degree m.
f) The number of edges of Km,n = mn. This is because the total degree of
Km,n is 2mn, and so, by Theorem 11.1.1, Km,n has 2mn = mn edges.
2
Section 10.1 #37
Find which of the following graphs are bipartite. Redraw
the bipartite graphs so that their bipartite nature is evident.
Solution
a.)
v1
v2
v3
v4
b.)Not bipartite. No possible way to create two subsets without vertices relating to one
another within the same subset.
c.)
v1
v2
v3
v4
v5
v6
d.) Not bipartite. No possible way to create two subsets without vertices relating to one
another within the same subset.
Solution Continued
d.) Not bipartite. No possible way to create two subsets without vertices relating
to one another within the same subset.
e.)
v1
v2
v3
v5
v4
f.) Not bipartite. No possible way to create two subsets without vertices relating to
one another within the same subset.
Summary of Main Results
 Definition: Let n be a positive integer. A complete
graph on n vertices, denoted Kn, is a simple graph
with n vertices and exactly one edge connecting each
pair of distinct vertices.
 Definition: Let m and n be positive integers. A complete
bipartite graph on (m,n) vertices, denoted Km,n, is a
simple graph with distinct vertices
v1, v2, … ,vm and w1, w2, … ,wn that satisfies the
following properties: For all i,k = 1, 2, … ,m and for all
j, l = 1, 2, … , n,
1. There is an edge from each vertex vi to each vertex wj.
2. There is no edge from any vertex vi to any other vertex
vk.
3. There is no edge from any vertex wj to any other vertex
wl .
Leonhard Euler
Euler lived from April 15, 1707 to
September 18, 1783. Euler is
considered one of the greatest
mathematician of all time. With
contributions to calculus and
wrote the first paper on graph
theory. Not only was he a great
mathematician but, he was also
renowned for his work in
mechanics, fluid dynamics,
optics, and astronomy.
```