4. All graphs in this question are finite and simple. (a) If I is a tree with at least 2 vertices, show that x(T) = 2. (b) Determine x(Cn, k), the number of k colourings of the cycle Cn with n vertices, for n> 3. In parts (c) and (d) below, G is a graph such that x(G) > x(G - v) for all v € V(G). That is, for any v E V(G) the graph G-v has a colouring with less colours than G does. (c) Show that G is connected. (d) Show that dc(v) > x(G) – 1 for all v € V(G), that is, all vertices of G have degree at least X(G) - 1.
4. All graphs in this question are finite and simple. (a) If I is a tree with at least 2 vertices, show that x(T) = 2. (b) Determine x(Cn, k), the number of k colourings of the cycle Cn with n vertices, for n> 3. In parts (c) and (d) below, G is a graph such that x(G) > x(G - v) for all v € V(G). That is, for any v E V(G) the graph G-v has a colouring with less colours than G does. (c) Show that G is connected. (d) Show that dc(v) > x(G) – 1 for all v € V(G), that is, all vertices of G have degree at least X(G) - 1.
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
c
d
![4. All graphs in this question are finite and simple.
(a) If I is a tree with at least 2 vertices, show that x(T) = 2.
(b) Determine x(Cn, k), the number of k colourings of the cycle Cn with n vertices, for
n> 3.
In parts (c) and (d) below, G is a graph such that x(G) > x(G - v) for all v € V(G).
That is, for any v E V(G) the graph G-v has a colouring with less colours than G does.
(c) Show that G is connected.
(d) Show that dc(v) > x(G) – 1 for all v € V(G), that is, all vertices of G have degree at
least X(G) - 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a4af2c9-a3ce-409e-892c-ae7106d06bb0%2Fed722a4b-2342-4924-b880-474e2371cce3%2Fu04xk9s_processed.png&w=3840&q=75)
Transcribed Image Text:4. All graphs in this question are finite and simple.
(a) If I is a tree with at least 2 vertices, show that x(T) = 2.
(b) Determine x(Cn, k), the number of k colourings of the cycle Cn with n vertices, for
n> 3.
In parts (c) and (d) below, G is a graph such that x(G) > x(G - v) for all v € V(G).
That is, for any v E V(G) the graph G-v has a colouring with less colours than G does.
(c) Show that G is connected.
(d) Show that dc(v) > x(G) – 1 for all v € V(G), that is, all vertices of G have degree at
least X(G) - 1.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
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 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)