{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

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

View Full Document Right Arrow Icon
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
Image of page 1
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

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern