Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

4.2. Trees 253
are adjacent (they are siblings), so we are good so far. Now put
into A every child of every vertex in B (i.e., every grandchild of the
root). Keep going until all vertices have been assigned one of the
sets, alternating between A and B every “generation.” That is, a
vertex is in set B if and only if it is the child of a vertex in set A.
The key to how we partitioned the tree in the example was to know
which vertex to assign to a set next. We chose to visit all vertices in the
same generation before any vertices of the next generation. This is usually
called a breadth first search (we say “search” because you often traverse a
tree looking for vertices with certain properties).
In contrast, we could also have partitioned the tree in a different order.
Start with the root, put it in A. Then look for one child of the root to put in
B. Then find a child of that vertex, into A, and then find its child, into B,
and so on. When you get to a vertex with no children, retreat to its parent
and see if the parent has any other children. So we travel as far from the
root as fast as possible, then backtrack until we can move forward again.
This is called depth first search.
These algorithmic explanations can serve as a proof that every tree is
bipartite, although care needs to be spent to prove that the algorithms
are correct. Another approach to prove that all trees are bipartite, using
induction, is requested in the exercises.
Spanning Trees
One of the advantages of trees is that they give us a few simple ways to
travel through the vertices. If a connected graph is not a tree, then we can
still use these traversal algorithms if we identify a subgraph that is a tree.
First we should consider if this even makes sense. Given any connected
graph G, will there always be a subgraph that is a tree? Well, that is
actually too easy: you could just take a single vertex of G. If we want to
use this subgraph to tell us how to visit all vertices, then we want our
subgraph to include all of the vertices. We call such a tree a spanning tree.
It turns out that every connected graph has one (and usually many).
Spanning tree.
Given a connected graph G, a spanning tree of G is a subgraph of
G which is a tree and includes all the vertices of G.
Every connected graph has a spanning tree.
How do we know? We can give an algorithm for finding a spanning
tree! Start with a connected graph G. If there is no cycle, then G is already
a tree and we are done. If there is a cycle, let e be any edge in that cycle