By Nagpal, Radhika; Meyer, Albert R

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2. A partition of a set A is a collection of subsets {A1 , . . , Ak } such that any two of them are disjoint (for any i �= j, Ai ∩ Aj = ∅) and such that their union is A. Let R be an equivalence relation on the set A. For an element a ∈ A, let [a] denote the set {b ∈ A given a ∼R b}. We call this set the equivalence class of a under R. We call a a representative of [a]. 3. The sets [a] for a ∈ A constitute a partition of A. That is, for every a, b ∈ A, either [a] = [b] or [a] ∩ [b] = ∅. Proof.

If one directory contains another, there there is an edge between the associated internal nodes. If a directory contains a file, then there is an edge between the internal node and a leaf. In both family trees and directories, there can be exceptions that make the graph not a tree—relatives can marry and have children, while directories sometimes use soft links to create multiple directory entries for a given file. There are in fact many different equivalent ways of defining trees formally. ) Course Notes 4: Graphs 13 Figure 8: This tree has 11 leaves, which are defined as vertices of degree at most one.

In this case, p = 0 and the algorithm requires p + 1 = 1 colors. In the inductive step, assume P (n) to prove P (n + 1). Let G� be the graph obtained from G by removing vertex v and incident edges. No vertex in G� has degree greater than p, since removing a vertex and incident edges can not increase the degree of any other vertex. ) By induction, the algorithm (applied recursively) colors G� with at most p + 1 colors. Now we add back vertex v. Since v has at most p neighbors and there are p + 1 colors available, there is always one color left over for vertex v.

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6.042J / 18.062J Mathematics for Computer Science (SMA 5512), Fall 2002 by Nagpal, Radhika; Meyer, Albert R

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