Let G = (V, E) be a simple undirected graph (i.e. it has no self-loops or parallel edges). Use the Pigeonhole Principle to prove that every connected component of G with at least two vertices contains two vertices of the same degree. Note: We did examples of applying the PHP in lecture, drills, Tutorial 5 and PS3. Refer to these to help structure your solutions. Hint: When using the Pigeonhole Principle, always • clearly define your set A (of pigeons), • clearly define your set B (of pigeonholes), clearly define the function f : A → B that maps each pigeon a € A to a single pigeonhole f(a) and that f(a) € B (i.e. f has the 3 properties of a well-defined function), and explain how you're able to apply the Pigeonhole Principle (or its extended version) to obtain the desired result.
Let G = (V, E) be a simple undirected graph (i.e. it has no self-loops or parallel edges). Use the Pigeonhole Principle to prove that every connected component of G with at least two vertices contains two vertices of the same degree. Note: We did examples of applying the PHP in lecture, drills, Tutorial 5 and PS3. Refer to these to help structure your solutions. Hint: When using the Pigeonhole Principle, always • clearly define your set A (of pigeons), • clearly define your set B (of pigeonholes), clearly define the function f : A → B that maps each pigeon a € A to a single pigeonhole f(a) and that f(a) € B (i.e. f has the 3 properties of a well-defined function), and explain how you're able to apply the Pigeonhole Principle (or its extended version) to obtain the desired result.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let G = (V, E) be a simple undirected graph (i.e. it has no self-loops or parallel edges). Use
the Pigeonhole Principle to prove that every connected component of G with at least two vertices
contains two vertices of the same degree.
Note: We did examples of applying the PHP in lecture, drills, Tutorial 5 and PS3. Refer to these
to help structure your solutions.
Hint: When using the Pigeonhole Principle, always
• clearly define your set A (of pigeons),
• clearly define your set B (of pigeonholes),
clearly define the function f : A → B that maps each pigeon a € A to a single pigeonhole
f(a) and that f(a) € B (i.e. f has the 3 properties of a well-defined function), and
explain how you're able to apply the Pigeonhole Principle (or its extended version) to obtain
the desired result.
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