{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

T7 - mum number of edges G can have Problem 5 You are given...

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

View Full Document Right Arrow Icon
Problem 1: Suppose G is a tree with no vertices of degree 2. Prove that at least half of the vertices in G are leaves (i.e. have degree 1). Problem 2: Let G be a connected graph. Suppose the algorithm of growing a breadth first search tree produces the same tree no matter which vertex in G was chosen as the root. Prove that G is a tree. Problem 3: Let G be a graph with p vertices, q edges and minimum vertex degree k . a) Show that the number of triangles (cycles of length 3) is at least q (2 k - p ) 3 . (In particular, if p 2 < k , then the graph contains a triangle.) b) Find a graph which achieves this bound for every p . Problem 4: Suppose G is a bipartite graph on p vertices. What is the maxi-
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: mum number of edges G can have? Problem 5: You are given a standard 8 by 8 chessboard where each square has length 1. Is it possible to place 2 by 1 paper rectangles on the board so that all of the squares are covered except for two opposite corners? (Rectangles must be placed so they cover exactly two squares of the board and do not overlap) Demonstrate a valid tiling or prove none exists. Problem 6: a) Prove that every graph on 6 vertices either contains a 3-cycle or a set of 3 vertices that are pairwise non-adjacent; b) Show that the above statement is not true for graphs on 5 vertices. 1...
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