text: Explore BrainMass
  
  # Euler Digraph: Characterization
  
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  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.
  
  https://brainmass.com/math/graphs-and-functions/euler-digraph-characterization-122201
  
  #### 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 ...
  
  #### 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.
  
  \$2.49
explanation: The provided text is an excerpt from a solution to a set of graph theory problems, specifically dealing with Eulerian and Hamiltonian properties of digraphs. It does not contain a direct question but rather a detailed explanation and solution to the problems mentioned. The text explains the solutions to three problems: a modified version of the Konigsberg bridge problem, a Hamiltonian cycle in a dodecahedron, and a theorem about Eulerian digraphs.
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distilabel_metadata.content: Explore BrainMass
  
  # Euler Digraph: Characterization
  
  Not what you're looking for? Search our solutions OR ask your own Custom question.
  
  This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
  
  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.
  
  https://brainmass.com/math/graphs-and-functions/euler-digraph-characterization-122201
  
  #### 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 ...
  
  #### 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.
  
  \$2.49
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an excerpt from a solution to a set of graph theory problems, specifically dealing with Eulerian and Hamiltonian properties of digraphs. It does not contain a direct question but rather a detailed explanation and solution to the problems mentioned. The text explains the solutions to three problems: a modified version of the Konigsberg bridge problem, a Hamiltonian cycle in a dodecahedron, and a theorem about Eulerian digraphs.
  
  Label: no-question