Euler graph and Eulerian path
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is it always true that if a graph contains a Eulerian path(at least one) then it will always be a Euler graph?
I know that all the vertices of a graph should be of even degree to become a Euler graph as well as the graph should be connected too, but was having this self-doubt.
EDIT:
what about this graph:
Euler or not ?
graph-theory
asked Dec 29 '18 at 13:02
ashwani yadavashwani yadav
254
$endgroup$
add a comment |
$begingroup$
is it always true that if a graph contains a Eulerian path(at least one) then it will always be a Euler graph?
I know that all the vertices of a graph should be of even degree to become a Euler graph as well as the graph should be connected too, but was having this self-doubt.
EDIT:
what about this graph:
Euler or not ?
graph-theory
asked Dec 29 '18 at 13:02
ashwani yadavashwani yadav
254
$endgroup$
$begingroup$
that's the doubt .... first i was going with this definition -" a euler graph should contain a euler path i.e should be able to walk on all edges without repeating any edge " then i found some graphs which contains an eulerian path but they were not euler graph as all vertices was not having even degree.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:08
$begingroup$
A straight line graph has an obvious Euler path, for instance, but no cycles at all. Even the graph with two vertices connected by a single edge has an Euler path.
$endgroup$
– lulu
Dec 29 '18 at 13:12
add a comment |
$begingroup$
is it always true that if a graph contains a Eulerian path(at least one) then it will always be a Euler graph?
I know that all the vertices of a graph should be of even degree to become a Euler graph as well as the graph should be connected too, but was having this self-doubt.
EDIT:
what about this graph:
Euler or not ?
graph-theory
asked Dec 29 '18 at 13:02
ashwani yadavashwani yadav
254
$endgroup$
is it always true that if a graph contains a Eulerian path(at least one) then it will always be a Euler graph?
I know that all the vertices of a graph should be of even degree to become a Euler graph as well as the graph should be connected too, but was having this self-doubt.
EDIT:
what about this graph:
Euler or not ?
graph-theory
graph-theory
asked Dec 29 '18 at 13:02
ashwani yadavashwani yadav
254
asked Dec 29 '18 at 13:02
ashwani yadavashwani yadav
254
edited Dec 29 '18 at 13:31
ashwani yadav
asked Dec 29 '18 at 13:02
ashwani yadavashwani yadav
254
asked Dec 29 '18 at 13:02
ashwani yadavashwani yadav
254
asked Dec 29 '18 at 13:02
ashwani yadavashwani yadav
254
254
$begingroup$
that's the doubt .... first i was going with this definition -" a euler graph should contain a euler path i.e should be able to walk on all edges without repeating any edge " then i found some graphs which contains an eulerian path but they were not euler graph as all vertices was not having even degree.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:08
$begingroup$
A straight line graph has an obvious Euler path, for instance, but no cycles at all. Even the graph with two vertices connected by a single edge has an Euler path.
$endgroup$
– lulu
Dec 29 '18 at 13:12
add a comment |
$begingroup$
that's the doubt .... first i was going with this definition -" a euler graph should contain a euler path i.e should be able to walk on all edges without repeating any edge " then i found some graphs which contains an eulerian path but they were not euler graph as all vertices was not having even degree.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:08
$begingroup$
A straight line graph has an obvious Euler path, for instance, but no cycles at all. Even the graph with two vertices connected by a single edge has an Euler path.
$endgroup$
– lulu
Dec 29 '18 at 13:12
$begingroup$
that's the doubt .... first i was going with this definition -" a euler graph should contain a euler path i.e should be able to walk on all edges without repeating any edge " then i found some graphs which contains an eulerian path but they were not euler graph as all vertices was not having even degree.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:08
$begingroup$
that's the doubt .... first i was going with this definition -" a euler graph should contain a euler path i.e should be able to walk on all edges without repeating any edge " then i found some graphs which contains an eulerian path but they were not euler graph as all vertices was not having even degree.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:08
$begingroup$
A straight line graph has an obvious Euler path, for instance, but no cycles at all. Even the graph with two vertices connected by a single edge has an Euler path.
$endgroup$
– lulu
Dec 29 '18 at 13:12
$begingroup$
A straight line graph has an obvious Euler path, for instance, but no cycles at all. Even the graph with two vertices connected by a single edge has an Euler path.
$endgroup$
– lulu
Dec 29 '18 at 13:12
add a comment |
1 Answer
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According to Wolfram Mathworld an Euler graph is a graph containing an Eulerian cycle. There surely are examples of graphs with an Eulerian path, but not an Eulerian cycle. Consider two connected vertices for example.
EDIT: The link also mentions some authors define an Euler graph as a connected graph where every vertex has even degree. It is by no means trivial to show equivalence between these definition, but one of the two implications is pretty easy. If a graph has a Eulerian cycle, then every vertex must be entered and left an equal amount of times in the cycle. Since every edge can only be visited once, we find an even amount of edges per vertex. ($2$ times the amount of times the vertex is visited in the cycle)
answered Dec 29 '18 at 13:05
SmileyCraftSmileyCraft
3,749519
$endgroup$
$begingroup$
edited the question, explain with that graph -Euler or not.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:33
$begingroup$
It has an Eulerian path, but not an Eulerian cycle. Hence, accordint to Wolfram Mathworld the graph is not an Euler graph.
$endgroup$
– SmileyCraft
Dec 29 '18 at 15:06
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
According to Wolfram Mathworld an Euler graph is a graph containing an Eulerian cycle. There surely are examples of graphs with an Eulerian path, but not an Eulerian cycle. Consider two connected vertices for example.
EDIT: The link also mentions some authors define an Euler graph as a connected graph where every vertex has even degree. It is by no means trivial to show equivalence between these definition, but one of the two implications is pretty easy. If a graph has a Eulerian cycle, then every vertex must be entered and left an equal amount of times in the cycle. Since every edge can only be visited once, we find an even amount of edges per vertex. ($2$ times the amount of times the vertex is visited in the cycle)
answered Dec 29 '18 at 13:05
SmileyCraftSmileyCraft
3,749519
$endgroup$
$begingroup$
edited the question, explain with that graph -Euler or not.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:33
$begingroup$
It has an Eulerian path, but not an Eulerian cycle. Hence, accordint to Wolfram Mathworld the graph is not an Euler graph.
$endgroup$
– SmileyCraft
Dec 29 '18 at 15:06
add a comment |
$begingroup$
According to Wolfram Mathworld an Euler graph is a graph containing an Eulerian cycle. There surely are examples of graphs with an Eulerian path, but not an Eulerian cycle. Consider two connected vertices for example.
EDIT: The link also mentions some authors define an Euler graph as a connected graph where every vertex has even degree. It is by no means trivial to show equivalence between these definition, but one of the two implications is pretty easy. If a graph has a Eulerian cycle, then every vertex must be entered and left an equal amount of times in the cycle. Since every edge can only be visited once, we find an even amount of edges per vertex. ($2$ times the amount of times the vertex is visited in the cycle)
answered Dec 29 '18 at 13:05
SmileyCraftSmileyCraft
3,749519
$endgroup$
$begingroup$
edited the question, explain with that graph -Euler or not.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:33
$begingroup$
It has an Eulerian path, but not an Eulerian cycle. Hence, accordint to Wolfram Mathworld the graph is not an Euler graph.
$endgroup$
– SmileyCraft
Dec 29 '18 at 15:06
add a comment |
$begingroup$
According to Wolfram Mathworld an Euler graph is a graph containing an Eulerian cycle. There surely are examples of graphs with an Eulerian path, but not an Eulerian cycle. Consider two connected vertices for example.
EDIT: The link also mentions some authors define an Euler graph as a connected graph where every vertex has even degree. It is by no means trivial to show equivalence between these definition, but one of the two implications is pretty easy. If a graph has a Eulerian cycle, then every vertex must be entered and left an equal amount of times in the cycle. Since every edge can only be visited once, we find an even amount of edges per vertex. ($2$ times the amount of times the vertex is visited in the cycle)
answered Dec 29 '18 at 13:05
SmileyCraftSmileyCraft
3,749519
$endgroup$
According to Wolfram Mathworld an Euler graph is a graph containing an Eulerian cycle. There surely are examples of graphs with an Eulerian path, but not an Eulerian cycle. Consider two connected vertices for example.
EDIT: The link also mentions some authors define an Euler graph as a connected graph where every vertex has even degree. It is by no means trivial to show equivalence between these definition, but one of the two implications is pretty easy. If a graph has a Eulerian cycle, then every vertex must be entered and left an equal amount of times in the cycle. Since every edge can only be visited once, we find an even amount of edges per vertex. ($2$ times the amount of times the vertex is visited in the cycle)
answered Dec 29 '18 at 13:05
SmileyCraftSmileyCraft
3,749519
edited Dec 29 '18 at 13:12
answered Dec 29 '18 at 13:05
SmileyCraftSmileyCraft
3,749519
answered Dec 29 '18 at 13:05
SmileyCraftSmileyCraft
3,749519
answered Dec 29 '18 at 13:05
SmileyCraftSmileyCraft
3,749519
3,749519
$begingroup$
edited the question, explain with that graph -Euler or not.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:33
$begingroup$
It has an Eulerian path, but not an Eulerian cycle. Hence, accordint to Wolfram Mathworld the graph is not an Euler graph.
$endgroup$
– SmileyCraft
Dec 29 '18 at 15:06
add a comment |
$begingroup$
edited the question, explain with that graph -Euler or not.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:33
$begingroup$
It has an Eulerian path, but not an Eulerian cycle. Hence, accordint to Wolfram Mathworld the graph is not an Euler graph.
$endgroup$
– SmileyCraft
Dec 29 '18 at 15:06
$begingroup$
edited the question, explain with that graph -Euler or not.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:33
$begingroup$
edited the question, explain with that graph -Euler or not.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:33
$begingroup$
It has an Eulerian path, but not an Eulerian cycle. Hence, accordint to Wolfram Mathworld the graph is not an Euler graph.
$endgroup$
– SmileyCraft
Dec 29 '18 at 15:06
$begingroup$
It has an Eulerian path, but not an Eulerian cycle. Hence, accordint to Wolfram Mathworld the graph is not an Euler graph.
$endgroup$
– SmileyCraft
Dec 29 '18 at 15:06
add a comment |
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$begingroup$
that's the doubt .... first i was going with this definition -" a euler graph should contain a euler path i.e should be able to walk on all edges without repeating any edge " then i found some graphs which contains an eulerian path but they were not euler graph as all vertices was not having even degree.
$endgroup$
– ashwani yadav
Dec 29 '18 at 13:08
$begingroup$
A straight line graph has an obvious Euler path, for instance, but no cycles at all. Even the graph with two vertices connected by a single edge has an Euler path.
$endgroup$
– lulu
Dec 29 '18 at 13:12