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
icon
Related questions
Question

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.

Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 3 images

Blurred answer
Knowledge Booster
Paths and Circuits
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,