A7 - E from G so that each component of the remaining graph...

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

View Full Document Right Arrow Icon
MATH 239 Spring 2010 Assignment 7 Due: July 9, 2010, 11:45AM EDT 1. (a) (3 marks) For p 3 , prove that a connected graph with p vertices and p edges contains exactly one cycle. (b) (3 marks) Determine the set of all graphs G with at least 3 vertices which have the property that G - e is a tree for each edge e in G . (In justifying your answer, you need to say why these are the only graphs with this property.) 2. (6 marks) Suppose that T is a tree containing only odd-degree vertices where each ver- tex has degree at most 7, and T has exactly 10 leaves. (a) What is the maximum number of vertices that T can have? Draw an example of T with this many vertices. (b) What is the minimum number of vertices that T must have? Draw an example of T with this many vertices. 3. (5 marks) Let G be a graph whose vertices have degrees either 1 or 3. Let S be the set of vertices of degree 1. Suppose that we can remove a set of edges
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: E from G so that each component of the remaining graph is a tree containing exactly one vertex in S . Prove that |E| = 1 2 | V ( G ) | . 4. (8 marks) Consider the following graph G . 1 2 3 4 5 6 7 8 9 10 11 12 13 (a) Construct a breadth-first search tree for G , taking the vertex labelled 1 as root. When considering the vertices adjacent to the vertex being examined, take them in increasing order of their labels. (b) Determine the set of vertices whose distance to vertex 1 is two. (c) Determine whether or not G is bipartite, and prove your assertion. 5. (5 marks) Suppose that a connected graph G has a breadth-first search tree T for which every non-tree edge joins vertices at equal levels. Prove that every cycle of G contains an even number of tree edges....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online