Problem 1 Recall from Tutorial 8 that a subset of vertices S CV is an independent set in G = (V, E) if and only if Vu, v € S : {u, v} & E (that is, no two vertices in S are adjacent to each other.) Use the extended Pigeonhole Principle to show that if G = (V, E) is k-colourable then there exists an independent set in G of size at least [2], where n = |VI.
Problem 1 Recall from Tutorial 8 that a subset of vertices S CV is an independent set in G = (V, E) if and only if Vu, v € S : {u, v} & E (that is, no two vertices in S are adjacent to each other.) Use the extended Pigeonhole Principle to show that if G = (V, E) is k-colourable then there exists an independent set in G of size at least [2], where n = |VI.
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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![Problem 1
Recall from Tutorial 8 that a subset of vertices S CV is an independent set in G = (V, E) if and
only if Vu, v € S : {u, v} & E (that is, no two vertices in S are adjacent to each other.)
Use the extended Pigeonhole Principle to show that if G = (V, E) is k-colourable then there exists
an independent set in G of size at least [2], where n = |VI.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20fe110a-296a-48a1-a7d8-d1b56fc22b3f%2F81e03437-2610-45a4-b0ac-95de84f248f4%2Fof2oyk_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1
Recall from Tutorial 8 that a subset of vertices S CV is an independent set in G = (V, E) if and
only if Vu, v € S : {u, v} & E (that is, no two vertices in S are adjacent to each other.)
Use the extended Pigeonhole Principle to show that if G = (V, E) is k-colourable then there exists
an independent set in G of size at least [2], where n = |VI.
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