MATH 443
Assignment #3
Due Tuesday November 5.
1. Let G be a simple 3-regular graph. Recall that P4 refers to the path of three edges (and 4
vertices).
a) Prove that if G has a (edge) decomposition into P4 ’s then G has a perfect matching.
b) Prove that if G has a perfect matching then G has a (edge) decomposition into P4 ’s.
2. Prove that a tree T has a perfect matching if and only if for each vertex v ∈ V (T ) that
o(T \v) = 1 where o(T \v) is the number of odd components of T \v.
3. Find a simple planar graph that has precisely 2 vertices of degree 6 and all the rest have
degree 5. Icosahedron?
4. Show that a simple plane graph with no faces of size 3 or less must have at most |V (G)| − 2
faces. (for example the cube Q3 ).
5. Consider an arbitrary drawing of Kn in the plane. Necessarily it will have edge crossings
(for
1 n
n ≥ 5). Prove
that the numbers of pairs of edges that must cross is at least 5 4 . Hmm.
n
1 n
1
=
.
5 4
n−4 5