UMBC CMSC203 Discrete Structures, Section 06, Spring 2016
Homework 2, Due Thursday, 02/11
Vertex Cover.
Consider a graph G. Let X be a subset of the vertices
in G. We say that X is a vertex cover of
G if for every edge in G at least one of the endpoints
of that edge is in X.
In the graph below, find a vertex cover with as few vertices as you can.
List the vertices of the vertex cover you found and briefly argue that it
is the smallest possible.
Hint: the smallest vertex cover in this graph has 10 vertices.
Hamiltonian Circuits.
[Adapted from Problem Solving Through Recreational Mathematics by
Averbach & Chein, 1980.]
A graph has a Hamiltonian circuit if there is a path in the
graph that visits every vertex exactly once and returns to the first vertex
in the path.
Does the graph below have a Hamiltonian circuit? Justify your answer.
In general, if a graph has a Hamiltonian circuit, is it necessarily the
case that the graph has an Euler circuit? Justify your answer.
In general, if a graph has an Euler circuit, is it necessarily the
case that the graph has a Hamiltonian circuit? Justify your answer.
Regular Graphs.
In a d-regular graph, every vertex in the graph has degree d.
Recall that the degree of a vertex is the number of edges incident on the
vertex. (I.e., count the number of edges coming out of a vertex and that is
its degree.)
Draw a 3-regular graph with 6 vertices.
Draw a 3-regular graph with 8 vertices.
Draw a 3-regular graph with 10 vertices.
Are there any 3-regular graphs with 9 vertices?
why or why not?