notes-100107 - MATH 682 1 Notes Combinatorics and Graph...

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MATH 682 Notes Combinatorics and Graph Theory II 1 Trees Recall that we finished the last semester with the introduction of connectedness and connected com- ponents. This leads us to an interesting and useful class of graphs with several interesting properties. Definition 1. A connected graph G which does not contain any cycles is a tree . Among other things, this explains why we call an acyclic graph a forest — it’s a graph whose com- ponents are trees! Below the three distinct trees on five nodes are pictured: note that they look somewhat “tree-like”, visually speaking: We will prove an important property of trees, which will give us an extremely useful tool for building inductive proofs about trees. Proposition 1. If T is a tree with at least 2 vertices, then δ ( T ) = 1 Proof. Note that since | T | ≥ 2, every vertex in T has degree of at least 1, since a vertex of degree zero would not be connected to other vertices. We shall show that not every vertex can not have degree 2. Suppose that δ ( T ) 2. Then, choose a vertex u 0 in T . Since d ( u 0 ) 1, u 0 has a neighbor u 1 . Since d ( u 1 ) 2, u 1 has at least two neighbors: u 0 and some vertex u 2 6 = u 0 . Likewise, u 2 has u 1 and some distinct u 3 as neighbors. We may continue this process for as long as desired to get a walk u 0 u 1 u 2 u 3 ∼ · · · ∼ u k such that every three consecutive vertices are distinct for arbitrarily large k . By making k sufficiently large, since T is finite, this walk is guaranteed to self-intersect. Let j be the first index of self-intersection, so that for some i < j , u i = u j . By our distinctness criterion, i 6 = j - 1 and i 6 = j - 2, so i j - 3. Then, u i u i +1 ∼ · · · ∼ u j - 1 is a path of length at least 2, and thus u i u i +1 ∼ · · · ∼ u j - 1 u j = u i is a cycle; since T cannot contain a cycle, our assumption that δ ( T ) 2 must be erroneous. Definition 2. A vertex of degree 1 in a tree is called a leaf . The above proposition, then, shows that every tree has at least one leaf. It’s in fact fairly each to modify the proposition to show that every tree has at least two leaves, but not entirely necessary. One useful property of leaves is that they can be pruned: Theorem 1. If v is a leaf of tree T , then T - v is also a tree.
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