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Find any Hamiltonian circuit on the graph above. Give your answer as a list of vertices, starting and ending at the same vertex. Example: ABCA?
connecting the vertices in a graph so that the route traveled starts and ends at the same vertex.
What else can I help you with?
What is the difference between a Hamiltonian circuit and a Euler circuit?
In an Euler circuit we go through the whole circuit without picking the pencil up. In doing so, the edges can never be repeated but vertices may repeat. In a Hamiltonian circuit the vertices and edges both can not repeat. So Avery Hamiltonain circuit is also Eulerian but it is not necessary that every euler is also Hamiltonian.
What is a Hamiltonian Circuit?
a path that starts and ends at the same vertex and passes through all the other vertices exactly once...
Do all vertices have to be used once in Hamilton circuits?
Yes, in a Hamiltonian circuit, all vertices of a graph must be visited exactly once before returning to the starting vertex. This is a defining characteristic of Hamiltonian circuits, distinguishing them from other types of paths or circuits in graph theory, which may not require visiting all vertices. The aim is to create a closed loop that includes every vertex without repetition.
What is a hamiltonian path in a graph?
A Hamiltonian path in a graph is a path that visits every vertex exactly once. It does not need to visit every edge, only every vertex. If a Hamiltonian path exists in a graph, the graph is called a Hamiltonian graph.
Can a graph have a euler circuit but not a hamiltonian circuit?
Yes. Example: .................................................... ...A * ........................................... ......|.\ ......................................... eg Euler circuit: ACDCBA ......|...\ ........... --------- ............. ......|.....\........./...............\............ The Hamilton circuit is impossible as it has two ......|.......\...../...................\.......... halves (ACD & CD) connected to each other only ......|.........\./.......................\........ at vertex C. Once vertex C has been reached in ......|.......C *........................* D.... one half, it can only be used to start a path in ......|........./.\......................./......... the other half, or complete the cycle in the ......|......./.....\.................../........... current half; or if the path starts at C, it will end ......|...../.........\.............../............. without the other half being visited before C is ......|.../ ........... --------- .............. revisited. ......|./ ........................................... ...B *.............................................. ......................................................
Can a graph have an Euler circuit but not a Hamiltonian circuit?
Yes. An example: _____A---------B________ A connected directly to B and D by one path. _____|_______/|\________ B connected directly to A and E by one path, and to C by two paths. _____|______/_|_\_______ _____|_____/___\_|______ _____|__E/_____\|______ E connected directly to B and D by one path. _____|____\_____C______ C connected directly to B and D by two paths. _____|_____\____|\_____ _____|______\___|__\___ _____|_______\__|__/___ _____|________\_|_/____ _____|_________\|/_____ _____-------------D_____ D connected directly to A and E by one path, and to C by two paths. There is an Euler circuit: ABCDEBCDA But a Hamiltonian circuit is impossible: as part of a circuit A can only be reached by the path BAD, but once BAD has been traversed it is impossible to get to both C and E without returning to B or D first. However there is a Hamiltonian Path: ABCDE.
How do you eulerize a graph?
To eulerize a graph, you need to ensure that all vertices have even degrees, as this is a requirement for a graph to have an Eulerian circuit. If any vertices have odd degrees, you can add edges between pairs of odd-degree vertices to make their degrees even. The added edges can be chosen carefully to minimize the total length of the resulting Eulerian circuit. Finally, the resulting graph will have all vertices with even degrees, allowing for an Eulerian path or circuit.
What is an even vertices?
In graph theory, even vertices refer to vertices that have an even degree, meaning they are connected to an even number of edges. This property is significant in various concepts, such as Eulerian paths and circuits, where a graph can have an Eulerian circuit if all vertices have even degrees. Analyzing even vertices helps in understanding the structure and properties of graphs.
Short circuit example?
An example of a kind of short circuit is an arc welding.
What is the definition of circuit in math?
Circuit is a term often used in graph theory. Here is how it is defined: A simple circuit on n vertices, Cn is a connected graph with n vertices x1, x2,..., xn, each of which has degree 2, with xi adjacent to xi+1 for i=1,2,...,n-1 and xn adjacent to x1. Simple means no loops or multiple edges.
A toaster is an example of which type of circuit?
It is a resistive type of circuit.
What is an example of a circuit interrupter?
A ground fault circuit interrupter (GFCI) is an example of a circuit interrupter. It is designed to quickly shut off power in the event of a ground fault, which helps prevent electric shocks and fires in electrical circuits.
