Determine if graph 3 has an Euler circuit, an Euler path but not an Euler circui
ID: 3672681 • Letter: D
Question
Determine if graph 3 has an Euler circuit, an Euler path but not an Euler circuit, or neither. If the graph has a circuit or path, list the vertices in the circuit or path. If a circuit or path does not exist, give a brief explanation why it doesn't exist. A) B) Determine if graph 3 has a Hamilton circuit, a Hamilton path but not a Hamilton circuit, or neither. If the graph has a circuit or path, list the vertices in the circuit or path. If a circuit or path does not exist, give a brief explanation why it doesn't exist.Explanation / Answer
1.
Euler’s Theorem 1
If a graph has any vertices of odd degree, then it CANNOT have an EULER CIRCUIT.
AND
If a graph is connected and every vertex has even degree, then it has AT LEAST ONE EULER CIRCUIT (usually more).
Euler’s Theorem 2
If a graph has more than 2 vertices of odd degree, then it CANNOT have an EULER PATH.
AND
If a graph is connected and has exactly 2 vertices of odd degree, then it has AT LEAST ONE EULER PATH (usually more). Any such path must start at one of the odd-degree vertices and end at the other.
Node B, D, F, H have odd degree so there is no Euler circuit as well as no Euler path.
2.
A Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once.
Hamiltonian Path exists: A B C E H F G D
There is no Hamiltonian Ciruit. E acts as an center to visit nodes from up then from down and come back again requires at least two time traversal for E. This contradicts to definition of Hamiltonian Circuit.
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