When does the complete bipartite graph K n,m have an Euler Trail(Path)?

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So I know that an Euler trail must have no more than two odd degree vertices.

So does this mean that either $n$ or $m$ must be odd? Or is it $n = m + 1$?

edited 20 hours ago

Especially Lime

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asked 20 hours ago

johntc121

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up vote
1
down vote

favorite

So I know that an Euler trail must have no more than two odd degree vertices.

So does this mean that either $n$ or $m$ must be odd? Or is it $n = m + 1$?

edited 20 hours ago

Especially Lime

20.9k22655

asked 20 hours ago

johntc121

194

add a comment |

up vote
1
down vote

favorite

So I know that an Euler trail must have no more than two odd degree vertices.

So does this mean that either $n$ or $m$ must be odd? Or is it $n = m + 1$?

edited 20 hours ago

Especially Lime

20.9k22655

asked 20 hours ago

johntc121

194

So I know that an Euler trail must have no more than two odd degree vertices.

So does this mean that either $n$ or $m$ must be odd? Or is it $n = m + 1$?

proof-verification graph-theory bipartite-graph eulerian-path

edited 20 hours ago

Especially Lime

20.9k22655

asked 20 hours ago

johntc121

194

edited 20 hours ago

Especially Lime

20.9k22655

asked 20 hours ago

johntc121

194

edited 20 hours ago

Especially Lime

20.9k22655

edited 20 hours ago

Especially Lime

20.9k22655

edited 20 hours ago

Especially Lime

20.9k22655

asked 20 hours ago

johntc121

194

asked 20 hours ago

johntc121

194

asked 20 hours ago

johntc121

194

add a comment |

1 Answer
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up vote
2
down vote

accepted

You're right that it has an Euler trail if and only if the number of odd-degree vertices is at most $2$.

In $K_{m,n}$ there are $m$ vertices of degree $n$ and $n$ vertices of degree $n$.

If $m,n$ are both even then all degrees are even so there is an Euler trail.

If exactly one of them, say $m$, is odd then there are $n$ vertices of odd degree. So when can there be an Euler trail in this case?

If $m,n$ are both odd then there are $m+n$ odd degree vertices. When does this give an Euler trail?

answered 20 hours ago

Especially Lime

20.9k22655

So does this mean that there are two possiblities for the odd occurence? Ie... n = m =1? and n =2 and m = odd number >= 1?
– johntc121
20 hours ago

@johntc121 yes, that's exactly right (of course you can also swap $m$ and $n$ in the second case).
– Especially Lime
20 hours ago

Awesome. Thank you so much
– johntc121
14 hours ago

add a comment |

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

up vote
2
down vote

accepted

You're right that it has an Euler trail if and only if the number of odd-degree vertices is at most $2$.

In $K_{m,n}$ there are $m$ vertices of degree $n$ and $n$ vertices of degree $n$.

If $m,n$ are both even then all degrees are even so there is an Euler trail.

If exactly one of them, say $m$, is odd then there are $n$ vertices of odd degree. So when can there be an Euler trail in this case?

If $m,n$ are both odd then there are $m+n$ odd degree vertices. When does this give an Euler trail?

answered 20 hours ago

Especially Lime

20.9k22655

So does this mean that there are two possiblities for the odd occurence? Ie... n = m =1? and n =2 and m = odd number >= 1?
– johntc121
20 hours ago

@johntc121 yes, that's exactly right (of course you can also swap $m$ and $n$ in the second case).
– Especially Lime
20 hours ago

Awesome. Thank you so much
– johntc121
14 hours ago

add a comment |

up vote
2
down vote

accepted

You're right that it has an Euler trail if and only if the number of odd-degree vertices is at most $2$.

In $K_{m,n}$ there are $m$ vertices of degree $n$ and $n$ vertices of degree $n$.

If $m,n$ are both even then all degrees are even so there is an Euler trail.

If exactly one of them, say $m$, is odd then there are $n$ vertices of odd degree. So when can there be an Euler trail in this case?

If $m,n$ are both odd then there are $m+n$ odd degree vertices. When does this give an Euler trail?

answered 20 hours ago

Especially Lime

20.9k22655

So does this mean that there are two possiblities for the odd occurence? Ie... n = m =1? and n =2 and m = odd number >= 1?
– johntc121
20 hours ago

@johntc121 yes, that's exactly right (of course you can also swap $m$ and $n$ in the second case).
– Especially Lime
20 hours ago

Awesome. Thank you so much
– johntc121
14 hours ago

add a comment |

up vote
2
down vote

accepted

You're right that it has an Euler trail if and only if the number of odd-degree vertices is at most $2$.

In $K_{m,n}$ there are $m$ vertices of degree $n$ and $n$ vertices of degree $n$.

If $m,n$ are both even then all degrees are even so there is an Euler trail.

If exactly one of them, say $m$, is odd then there are $n$ vertices of odd degree. So when can there be an Euler trail in this case?

If $m,n$ are both odd then there are $m+n$ odd degree vertices. When does this give an Euler trail?

answered 20 hours ago

Especially Lime

20.9k22655

You're right that it has an Euler trail if and only if the number of odd-degree vertices is at most $2$.

In $K_{m,n}$ there are $m$ vertices of degree $n$ and $n$ vertices of degree $n$.

If $m,n$ are both even then all degrees are even so there is an Euler trail.

If exactly one of them, say $m$, is odd then there are $n$ vertices of odd degree. So when can there be an Euler trail in this case?

If $m,n$ are both odd then there are $m+n$ odd degree vertices. When does this give an Euler trail?

answered 20 hours ago

Especially Lime

20.9k22655

answered 20 hours ago

Especially Lime

20.9k22655

answered 20 hours ago

Especially Lime

20.9k22655

answered 20 hours ago

Especially Lime

20.9k22655

So does this mean that there are two possiblities for the odd occurence? Ie... n = m =1? and n =2 and m = odd number >= 1?
– johntc121
20 hours ago

@johntc121 yes, that's exactly right (of course you can also swap $m$ and $n$ in the second case).
– Especially Lime
20 hours ago

Awesome. Thank you so much
– johntc121
14 hours ago

add a comment |

So does this mean that there are two possiblities for the odd occurence? Ie... n = m =1? and n =2 and m = odd number >= 1?
– johntc121
20 hours ago

@johntc121 yes, that's exactly right (of course you can also swap $m$ and $n$ in the second case).
– Especially Lime
20 hours ago

Awesome. Thank you so much
– johntc121
14 hours ago

So does this mean that there are two possiblities for the odd occurence? Ie... n = m =1? and n =2 and m = odd number >= 1?
– johntc121
20 hours ago

@johntc121 yes, that's exactly right (of course you can also swap $m$ and $n$ in the second case).
– Especially Lime
20 hours ago

Awesome. Thank you so much
– johntc121
14 hours ago

add a comment |

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