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.
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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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.
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