Graph theory problem and connected components
$begingroup$
In a connected graph $G$ where degree of every vertex $v$ is even, show that $Gsetminus v$ has at most $dfrac{1}{2}deg(v)$ connected components.
$Gsetminus v$ is the graph which is left after removing $v$ and all of its incident edges from $G$.
graph-theory eulerian-path
edited Dec 24 '18 at 14:43
Alex Ravsky
42.6k32383
asked Dec 24 '18 at 6:57
hussain sagarhussain sagar
908
$endgroup$
add a comment |
$begingroup$
In a connected graph $G$ where degree of every vertex $v$ is even, show that $Gsetminus v$ has at most $dfrac{1}{2}deg(v)$ connected components.
$Gsetminus v$ is the graph which is left after removing $v$ and all of its incident edges from $G$.
graph-theory eulerian-path
edited Dec 24 '18 at 14:43
Alex Ravsky
42.6k32383
asked Dec 24 '18 at 6:57
hussain sagarhussain sagar
908
$endgroup$
add a comment |
$begingroup$
In a connected graph $G$ where degree of every vertex $v$ is even, show that $Gsetminus v$ has at most $dfrac{1}{2}deg(v)$ connected components.
$Gsetminus v$ is the graph which is left after removing $v$ and all of its incident edges from $G$.
graph-theory eulerian-path
edited Dec 24 '18 at 14:43
Alex Ravsky
42.6k32383
asked Dec 24 '18 at 6:57
hussain sagarhussain sagar
908
$endgroup$
In a connected graph $G$ where degree of every vertex $v$ is even, show that $Gsetminus v$ has at most $dfrac{1}{2}deg(v)$ connected components.
$Gsetminus v$ is the graph which is left after removing $v$ and all of its incident edges from $G$.
graph-theory eulerian-path
graph-theory eulerian-path
edited Dec 24 '18 at 14:43
Alex Ravsky
42.6k32383
asked Dec 24 '18 at 6:57
hussain sagarhussain sagar
908
edited Dec 24 '18 at 14:43
Alex Ravsky
42.6k32383
asked Dec 24 '18 at 6:57
hussain sagarhussain sagar
908
edited Dec 24 '18 at 14:43
Alex Ravsky
42.6k32383
edited Dec 24 '18 at 14:43
Alex Ravsky
42.6k32383
edited Dec 24 '18 at 14:43
Alex Ravsky
42.6k32383
42.6k32383
asked Dec 24 '18 at 6:57
hussain sagarhussain sagar
908
asked Dec 24 '18 at 6:57
hussain sagarhussain sagar
908
asked Dec 24 '18 at 6:57
hussain sagarhussain sagar
908
908
add a comment |
add a comment |
1 Answer
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$begingroup$
The graph $G$ is Eulerian. Fix any Eulerian cycle on $G$. It is easy to see, when we remove $v$ from $G$, the cycle will be cut into at most $frac 12deg v$ parts, each of which is connected. Since each connected component of $Gsetminus{v}$ is a union of these parts, there are at most $frac 12deg v$ components.
answered Dec 24 '18 at 14:42
Alex RavskyAlex Ravsky
42.6k32383
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The graph $G$ is Eulerian. Fix any Eulerian cycle on $G$. It is easy to see, when we remove $v$ from $G$, the cycle will be cut into at most $frac 12deg v$ parts, each of which is connected. Since each connected component of $Gsetminus{v}$ is a union of these parts, there are at most $frac 12deg v$ components.
answered Dec 24 '18 at 14:42
Alex RavskyAlex Ravsky
42.6k32383
$endgroup$
add a comment |
$begingroup$
The graph $G$ is Eulerian. Fix any Eulerian cycle on $G$. It is easy to see, when we remove $v$ from $G$, the cycle will be cut into at most $frac 12deg v$ parts, each of which is connected. Since each connected component of $Gsetminus{v}$ is a union of these parts, there are at most $frac 12deg v$ components.
answered Dec 24 '18 at 14:42
Alex RavskyAlex Ravsky
42.6k32383
$endgroup$
add a comment |
$begingroup$
The graph $G$ is Eulerian. Fix any Eulerian cycle on $G$. It is easy to see, when we remove $v$ from $G$, the cycle will be cut into at most $frac 12deg v$ parts, each of which is connected. Since each connected component of $Gsetminus{v}$ is a union of these parts, there are at most $frac 12deg v$ components.
answered Dec 24 '18 at 14:42
Alex RavskyAlex Ravsky
42.6k32383
$endgroup$
The graph $G$ is Eulerian. Fix any Eulerian cycle on $G$. It is easy to see, when we remove $v$ from $G$, the cycle will be cut into at most $frac 12deg v$ parts, each of which is connected. Since each connected component of $Gsetminus{v}$ is a union of these parts, there are at most $frac 12deg v$ components.
answered Dec 24 '18 at 14:42
Alex RavskyAlex Ravsky
42.6k32383
answered Dec 24 '18 at 14:42
Alex RavskyAlex Ravsky
42.6k32383
answered Dec 24 '18 at 14:42
Alex RavskyAlex Ravsky
42.6k32383
answered Dec 24 '18 at 14:42
Alex RavskyAlex Ravsky
42.6k32383
42.6k32383
add a comment |
add a comment |
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