Lecture 5
Copyright c
Sue Geller 2006
This week’s topic is Hamiltonian graphs in which we want to hit every
vertex once without repeating a vertex, but it doesn’t matter what edges we
use. Indeed, in most graphs we won’t use all the edges. As is in the reading,
Alexander Hamilton started the study of Hamiltonian graphs by creating a
traveling salesmen game.
Since we are skipping Exercise 3.7, I include the proof of Dirac’s Theorem
here so that you can use it in your solutions to the assigned Problems.
Proof: Suppose that there are nvertices. After adding edges, we may assume
that there is a path of the form (v1, v2,...,vn), where v1and vnare not
adjacent. I claim that the degree condition implies that, if we look at all the
vertices adjacent to vn, there must be one, say vk, whose successor is adjacent
to v1. For v1is connected to at least n/2 vertices and vnis connected to
at least n/2 vertices but v1is not connected to vn. Therefore they both
must be connected to at least one vertex, vk. Then a Hamiltonian cycle is
(v1,...,vk, vn, vn−1,...,vk+1, v1).
Problem 3.7 hint: Change the problem into one for a graph constructed
from the information given.
On Problem 3.8 be sure to prove your characterization, i.e., a necessary
and sufficient condition that the graph have a path that is both Eulerian and
Hamiltonian.
1
