Euler Digraph: Characterization
Not what you're looking for?
Page 90, Theorem 4.6 ( To show that a digraph is hamiltonian if and only if for each vertex v, indegree (v) =outdegree (v)
Page 92 Exercise problem 4.1 ( A modified version of konigsberg problem where, two extra bridges are built. )
Page 93, Figure 4.5 need to show as Hamilton. (Show that dodecahedron is hamiltonian)
See attached file for full problem description.
Purchase this Solution
Solution Summary
Here we prove the Necessary and sufficient condition for a digraph to be Eulerian. It also solves three problems from Graphs and Digraphs concerning a modified Konigsberg problem and provide a Hamiltonian cycle for dodecahedron.
Solution Preview
Problem 4.1 and 4.5, I have attached figures.
Problem 4.1:
-------------
We construct a graph of the present day Konigsberg bridges and see that there are exactly two vertices (A and B) have odd degree. We know that every connected graph with exactly 2 odd vertices have an Eulerian tour starting and ending at these vertices. Hence we can device such a route passing through all the edges. The attached figure shows such a route.
Problem of figure 4.5
--------
We need to show the graph of dodecahedron is Hamiltonian. We do this by providing such a cycle explicitly. In the attached figure, follow the vertex sequence 1,2,3,...,19,20,1 which is such a Hamiltonian cycle.
Theorem 4.6
-------------
Theorem: Let D be a connected non-trivial digraph. D is Eulerian if and
only if id(v) = od(v) for every vertex. (id:in degree, od:out degree).
Proof:
A trail in a graph is a path without repeated edges.
Suppose D is Eulerian. Consider any ...
Purchase this Solution
Free BrainMass Quizzes
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Probability Quiz
Some questions on probability