{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

assgt5

# assgt5 - B n n ≥ 1(b Determine the number of edges in B n...

This preview shows page 1. Sign up to view the full content.

MATH 239 ASSIGNMENT 5 Due Friday, October 31, at NOON 1. Let A n , n 1, be the graph whose vertices are given by the n -element subsets of { 1 ,..., 2 n + 1 } , where two vertices are adjacent if and only if they are disjoint subsets. (For example, when n = 3, vertex { 2 , 3 , 6 } is adjacent to { 1 , 5 , 7 } and { 4 , 5 , 7 } , but not to { 1 , 3 , 5 } .) (a) Determine the number of vertices in A n , n 1. (b) Determine the number of edges in A n , n 1. (c) Determine the values of n 1 for which A n contains a triangle. 2. Let B n , n 2, be the graph whose vertices are given by the { 0 , 1 } -strings of length n , and two vertices are adjacent if and only if they differ in two consecutive positions. (For example, when n = 5, vertex 10110 is adjacent to 11010 and 10000, but not to 00110.) (a) Determine the number of vertices in
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: B n , n ≥ 1. (b) Determine the number of edges in B n , n ≥ 1. (c) Determine the values of n ≥ 1 for which B n is connected. 3. Suppose that G is a graph in which the minimum vertex degree is equal to d , for some d ≥ 2. (a) Prove that G has a path of length at least d . (b) Prove that G has a cycle of length at least d + 1. 4. If H has p vertices, p ≥ 1, and every vertex in H has degree greater than or equal to 1 2 p , prove that H is connected. 5. A quartic tree is a tree whose vertices have degrees 1 or 4 only. Prove that every quartic tree with m vertices of degree 4 has a total of 3 m + 2 vertices, for m ≥ 0....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern