5 Proofs in Graphs (a) On the axis from San Francisco traffic habits to Los Angeles traffic habits, Old California is more towards San Francisco: that is, civilized. In Old California, all roads were one way streets. Suppose Old California had n cities (n ≥ 2) such that for every pair of cities X and Y, either X had a road to Y or Y had a road to X. Prove that there existed a city which was reachable from every other city by traveling through at most 2 roads. [Hint: Induction] (b) Consider a connected graph G with n vertices which has exactly 2m vertices of odd degree, where m > 0. Prove that there are m walks that together cover all the edges of G (i.e., each Spring 2024, HW 2 edge of G occurs in exactly one of the m walks, and each of the walks should not contain any particular edge more than once). [Hint: In lecture, we have shown that a connected undirected graph has an Eulerian tour if and only if every vertex has even degree. This fact may be useful in the proof.] (c) Prove that any graph G is bipartite if and only if it has no tours of odd length. [Hint: In one of the directions, consider the lengths of paths starting from a given vertex.]
5 Proofs in Graphs (a) On the axis from San Francisco traffic habits to Los Angeles traffic habits, Old California is more towards San Francisco: that is, civilized. In Old California, all roads were one way streets. Suppose Old California had n cities (n ≥ 2) such that for every pair of cities X and Y, either X had a road to Y or Y had a road to X. Prove that there existed a city which was reachable from every other city by traveling through at most 2 roads. [Hint: Induction] (b) Consider a connected graph G with n vertices which has exactly 2m vertices of odd degree, where m > 0. Prove that there are m walks that together cover all the edges of G (i.e., each Spring 2024, HW 2 edge of G occurs in exactly one of the m walks, and each of the walks should not contain any particular edge more than once). [Hint: In lecture, we have shown that a connected undirected graph has an Eulerian tour if and only if every vertex has even degree. This fact may be useful in the proof.] (c) Prove that any graph G is bipartite if and only if it has no tours of odd length. [Hint: In one of the directions, consider the lengths of paths starting from a given vertex.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![5 Proofs in Graphs
(a) On the axis from San Francisco traffic habits to Los Angeles traffic habits, Old California is
more towards San Francisco: that is, civilized. In Old California, all roads were one way
streets. Suppose Old California had n cities (n ≥2) such that for every pair of cities X and Y,
either X had a road to Y or Y had a road to X.
Prove that there existed a city which was reachable from every other city by traveling through
at most 2 roads.
[Hint: Induction]
(b) Consider a connected graph G with n vertices which has exactly 2m vertices of odd degree,
where m > 0. Prove that there are m walks that together cover all the edges of G (i.e., each
$ 70, Spring 2024, HW 02
2
edge of G occurs in exactly one of the m walks, and each of the walks should not contain any
particular edge more than once).
[Hint: In lecture, we have shown that a connected undirected graph has an Eulerian tour if and
only if every vertex has even degree. This fact may be useful in the proof.]
(c) Prove that any graph G is bipartite if and only if it has no tours of odd length.
[Hint: In one of the directions, consider the lengths of paths starting from a given vertex.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9dab2332-d7a1-4dfe-b3bf-aa29f23ec53b%2Fa72f56eb-f3ed-4637-a8c6-9a20e6ecb0bd%2F4pz9u9h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5 Proofs in Graphs
(a) On the axis from San Francisco traffic habits to Los Angeles traffic habits, Old California is
more towards San Francisco: that is, civilized. In Old California, all roads were one way
streets. Suppose Old California had n cities (n ≥2) such that for every pair of cities X and Y,
either X had a road to Y or Y had a road to X.
Prove that there existed a city which was reachable from every other city by traveling through
at most 2 roads.
[Hint: Induction]
(b) Consider a connected graph G with n vertices which has exactly 2m vertices of odd degree,
where m > 0. Prove that there are m walks that together cover all the edges of G (i.e., each
$ 70, Spring 2024, HW 02
2
edge of G occurs in exactly one of the m walks, and each of the walks should not contain any
particular edge more than once).
[Hint: In lecture, we have shown that a connected undirected graph has an Eulerian tour if and
only if every vertex has even degree. This fact may be useful in the proof.]
(c) Prove that any graph G is bipartite if and only if it has no tours of odd length.
[Hint: In one of the directions, consider the lengths of paths starting from a given vertex.]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
