By a result of Landau (1953), we know that every tournament has a king (a vertex from which every vertex is reachable by a path of length at most 2). Let T be a tournament such that δ-(T) ≥ 1, that is, d-(v) ≥ 1 for all v ∈ V (T). 1. Show that if x is a king in T, then T has another king in N-(x). 2. Using the answer to the previous question, prove that T has at least 3 kings. 3. For each n ≥ 3, give a construction of a tournament T' with n vertices such that δ-(T') ≥ 1 and T' has exactly 3 kings.

By a result of Landau (1953), we know that every tournament has a king (a vertex from which every vertex is reachable by a path of length at most 2). Let T be a tournament such that δ-(T) ≥ 1, that is, d-(v) ≥ 1 for all v ∈ V (T). 1. Show that if x is a king in T, then T has another king in N-(x). 2. Using the answer to the previous question, prove that T has at least 3 kings. 3. For each n ≥ 3, give a construction of a tournament T' with n vertices such that δ-(T') ≥ 1 and T' has exactly 3 kings.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 74EQ

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