combinatorics_day_3.pdf - 21-301 Combinatorics Spring 2019...

21-301: Combinatorics Spring 2019
Lecture 3: Trees
Lecturer: John Mackey Date: January 18, 2019
1 Review
Hamiltonian Circuit A simple closed path that passes through each vertex in the graph exactly once. A
graph is Hamiltonian iff it has a polygon as a spanning subgraph.
Consider the two graphs below:
The graph on the left is the dodecahedron, which has a spanning polygon and is Hamiltonian.
The graph on the right is called the Peterson graph, which does not have a spanning polygon and is not
Hamiltonian.
2 Labeled Trees
For all following sections, these definitions will be helpful:
[n] The set of all positive integers up to and including n .
[n]
n - 2
The set of sequences of length n - 2, where the numbers in each sequence are chosen from [ n ].
Theorem 2.1. There are n
n - 2
different labeled trees on n vertices.
To help illustrate this theorem, consider labeling a 3 vertex graph with [3]. The only nonisomorphic tree
with 3 vertices is that of a straight line, and we can get a new labeling by switching the label on the center
vertex:
1

Lecture 3: Trees 2
There are 3 = 3
3 - 2
ways to do this. Therefore our theorem holds for trees with 3 vertices.
We can show that this theorem works for trees with 4 vertices as well. There are two such nonisomorphic trees:
There are 12 ways to label the first tree and 4 ways to label the second. Therefore there are 16 = 4
4 - 2
ways
to label a tree with 4 vertices.
There are three nonisomorphic trees on 5 vertices:
There are 60 ways to label the first tree, 5 ways to label the second, and 60 ways to label the third. Therefore
there are 125 = 5 5 - 2 ways to label a tree with 5 vertices.
3 Pr¨ufer Codes
The proof of Theorem 2.1 relies on an algorithm developed by H. Pr¨ufer that can assign any tree T a unique
name in [ n ]
n - 2
that characterizes the tree. This is called the tree’s Pr¨ufer code.