Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 514 — #522
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Chapter 12
Simple Graphs
Problem 12.61.
In this problem you will prove:
Theorem. A graph G is 2-colorable iff it contains no odd length closed walk.
As usual with “iff” assertions, the proof splits into two proofs: part (a) asks you
to prove that the left side of the “iff” implies the right side. The other problem parts
prove that the right side implies the left.
(a) Assume the left side and prove the right side. Three to five sentences should
suffice.
(b) Now assume the right side. As a first step toward proving the left side, explain
why we can focus on a single connected component H within G.
(c) As a second step, explain how to 2-color any tree.
(d) Choose any 2-coloring of a spanning tree T of H . Prove that H is 2-colorable
by showing that any edge not in T must also connect different-colored vertices.
Homework Problems
Problem 12.62.
Suppose D D .d 1 ; d 2 ; : : : ; d n / is a list of the vertex degrees of some n-vertex tree
T for n 2. That is, we assume the vertices of T are numbered, and d i > 0 is the
degree of the ith vertex of T .
(a) Explain why
nX
d i D 2.n 1/: (12.6)
iD1
(b) Prove conversely that if D is a sequence of positive integers satisfying equation
(12.6), then D is a list of the degrees of the vertices of some n-vertex tree.
Hint: Induction.
(c) Assume that D satisfies equation (12.6). Show that it is possible to partition
D into two sets S 1 ; S 2 such that the sum of the elements in each set is the same.
Hint: Trees are bipartite.
Problem 12.63.
Prove Corollary 12.9.12: If all edges in a finite weighted graph have distinct weights,
then the graph has a unique MST.