What graph has a Euler path but not Hamiltonian path?

The complete bipartite graph K2,4 has an Eulerian circuit, but is non-Hamiltonian (in fact, it doesn't even contain a Hamiltonian path). Any Hamiltonian path would alternate colors (and there's not enough blue vertices). Since every vertex has even degree, the graph has an Eulerian circuit.

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What is an Eulerian circuit but not a Hamiltonian circuit?

Important: An Eulerian circuit traverses every edge in a graph exactly once, but may repeat vertices, while a Hamiltonian circuit visits each vertex in a graph exactly once but may repeat edges.

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Is every Eulerian graph also Hamiltonian?

An Eulerian graph G (a connected graph in which every vertex has even degree) necessarily has an Euler tour, a closed walk passing through each edge of G exactly once. This tour corresponds to a Hamiltonian cycle in the line graph L(G), so the line graph of every Eulerian graph is Hamiltonian.

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Which graph does not have a Hamiltonian path?

A graph with a vertex of degree one cannot have a Hamilton circuit. Moreover, if a vertex in the graph has degree two, then both edges that are incident with this vertex must be part of any Hamilton circuit.

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Which of the graphs contains Euler path but not Euler?

EULER'S THEOREM

If a graph has exactly two odd vertices, then it has at least one Euler path, but no Euler circuit.

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Euler and Hamiltonian Graph Problem - Graph Theory - Discrete Mathematics

19 related questions found

What is an example of a graph which has an Euler circuit but not a Hamiltonian circuit?

The complete bipartite graph K2,4 has an Eulerian circuit, but is non-Hamiltonian (in fact, it doesn't even contain a Hamiltonian path). Any Hamiltonian path would alternate colors (and there's not enough blue vertices). Since every vertex has even degree, the graph has an Eulerian circuit.

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Does the graph of Königsberg have a Euler path?

Since the graph corresponding to historical Königsberg has four nodes of odd degree, it cannot have an Eulerian path. An alternative form of the problem asks for a path that traverses all bridges and also has the same starting and ending point. Such a walk is called an Eulerian circuit or an Euler tour.

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Do all graphs have a Hamiltonian path?

We observe that not every graph is Hamiltonian; for instance, it is clear that a dis- connected graph cannot contain any Hamiltonian cycle/path. There are also connected graphs that are not Hamiltonian. For example, if a connected graph has a a vertex of degree one, then it cannot be Hamiltonian.

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Does every graph have a Hamiltonian path?

Not all graphs have a Hamilton circuit or path. There is no way to tell just by looking at a graph if it has a Hamilton circuit or path like you can with an Euler circuit or path. You must do trial and error to determine this. By the way if a graph has a Hamilton circuit then it has a Hamilton path.

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How do you prove a graph does not have a Hamiltonian path?

The most natural way to prove a graph isn't Hamiltonian is to do a case by case analysis of possible paths, showing it doesn't work. For instance, in lecture we outlined the proof that if you remove a vertex from the Icosian graph, than the result isn't Hamiltonian.

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Can a graph have a Hamiltonian circuit but no Euler circuit?

Consider the complete graph on four vertices, as drawn below: The graph has a Hamiltonian circuit A → B → C → D → A which runs around the outside of the above diagram. However, all four vertices of have odd degree. Since has vertices of odd degree, does not have an Eulerian circuit.

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Can a graph be both Euler and Hamiltonian?

To set the record clear: Yes. A Path can be both Eularian and Hamiltonian. A Hamiltonian path is a spanning path, and an Eularian path goes through each edge exactly once.

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Does every graph with an Euler circuit has a Hamilton circuit?

Nope. They are quite different. A Hamiltonian graph is one which has a Hamiltonian cycle. A Hamiltonian cycle is a cycle that visits every vertex of the graph exactly once.

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Does every simple graph have an Euler circuit?

A graph has an Euler circuit if and only if the degree of every vertex is even. A graph has an Euler path if and only if there are at most two vertices with odd degree.

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What is the difference between Euler path and Hamiltonian path?

Euler or Hamilton Paths

An Euler path is a path that passes through every edge exactly once. If the euler path ends at the same vertex from which is has started it is called as Euler cycle. A Hamiltonian path is a path that passes through every vertex exactly once (NOT every edge).

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Is a Euler circuit always a Euler path?

An Euler Path is a path that goes through every edge of a graph exactly once An Euler Circuit is an Euler Path that begins and ends at the same vertex. Euler's Theorem: 1. If a graph has more than 2 vertices of odd degree then it has no Euler paths.

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Can a graph have a Hamilton path but no Hamilton circuit?

A Hamilton Circuit is a Hamilton Path that begins and ends at the same vertex. *Unlike Euler Paths and Circuits, there is no trick to tell if a graph has a Hamilton Path or Circuit. A Complete Graph is a graph where every pair of vertices is joined by an edge.

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Does the Petersen graph have a Hamiltonian path?

The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no Hamiltonian cycle.

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Does every bipartite graph have a Euler path?

Every bipartite graph has an Euler path. Every vertex of a bipartite graph has even degree. A graph is bipartite if and only if the sum of the degrees of all the vertices is even.

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Is K5 a Euler circuit?

The vertices of K5 all have even degree so an Eulerian circuit exists, namely the sequence of edges 1,5,8,10,4,2,9,7,6,3 .

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Is K5 7 a Euler path?

Solution. K 4 does not have an Euler path or circuit. K 5 has an Euler circuit (so also an Euler path). K 5 , 7 does not have an Euler path or circuit.

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Which of the following is not a type of graph * Hamiltonian Euler tree path?

Answer = D Explanation: If some closed walk in a graph contains all the edges then the walk is called Euler. 32) Which of the following is not a type of graph ? Answer = D Explanation:Path is a way from one node no another but not a graph.

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What is a Hamiltonian cycle but not a Eulerian cycle?

A Hamiltonian cycle in a graph is a cycle that visits every vertex at least once, and an Eulerian cycle is a cycle that visits every edge once. In general graphs, the problem of finding a Hamiltonian cycle is NP-hard, while finding an Eulerian cycle is solvable in polynomial time.

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Can we have a graph which is neither Eulerian nor Hamiltonian?

Some graphs lack both an Eulerian and a Hamiltonian cycle such as Star Graph. Star Graph contains no cycle.

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What is the difference between a Euler path and a Euler graph?

Euler Graph - A connected graph G is called an Euler graph, if there is a closed trail which includes every edge of the graph G. Euler Path - An Euler path is a path that uses every edge of a graph exactly once. An Euler path starts and ends at different vertices.

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