Prove that every k-linked graph G is (2k − 1)-connected
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So a graph is $k$-linked if $G$ has at least $2k$ vertices, and for $x_1,ldots,x_k,y_1,ldots,y_k$ that are 2k distinct vertices there are disjoint paths $P_1,ldots,P_k$ joining $x_i$ to $y_i$.
$G$ is $(2k-1)$-connected if size of $G$ is greater than $2k-1$ and $G$ has no separator of size less than $2k-1$.
I have tried proof by contradiction but seem to be stuck, I presume it is fairly easy and am getting frustrated that I cant figure it out??
graph-theory graph-connectivity
edited Nov 23 at 14:51
hardmath
28.6k94994
asked Nov 23 at 14:23
cxh
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up vote
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So a graph is $k$-linked if $G$ has at least $2k$ vertices, and for $x_1,ldots,x_k,y_1,ldots,y_k$ that are 2k distinct vertices there are disjoint paths $P_1,ldots,P_k$ joining $x_i$ to $y_i$.
$G$ is $(2k-1)$-connected if size of $G$ is greater than $2k-1$ and $G$ has no separator of size less than $2k-1$.
I have tried proof by contradiction but seem to be stuck, I presume it is fairly easy and am getting frustrated that I cant figure it out??
graph-theory graph-connectivity
edited Nov 23 at 14:51
hardmath
28.6k94994
asked Nov 23 at 14:23
cxh
1
1
Please use Latex/MathJax. Could you elaborate on your attempts so far?
– Stockfish
Nov 23 at 14:29
It seems reasonable to leave the title in plain ASCII here. I've added MathJax markup and $LaTeX$ syntax to the body. See this introduction to posting mathematical notation.
– hardmath
Nov 23 at 14:53
add a comment |
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
So a graph is $k$-linked if $G$ has at least $2k$ vertices, and for $x_1,ldots,x_k,y_1,ldots,y_k$ that are 2k distinct vertices there are disjoint paths $P_1,ldots,P_k$ joining $x_i$ to $y_i$.
$G$ is $(2k-1)$-connected if size of $G$ is greater than $2k-1$ and $G$ has no separator of size less than $2k-1$.
I have tried proof by contradiction but seem to be stuck, I presume it is fairly easy and am getting frustrated that I cant figure it out??
graph-theory graph-connectivity
edited Nov 23 at 14:51
hardmath
28.6k94994
asked Nov 23 at 14:23
cxh
1
So a graph is $k$-linked if $G$ has at least $2k$ vertices, and for $x_1,ldots,x_k,y_1,ldots,y_k$ that are 2k distinct vertices there are disjoint paths $P_1,ldots,P_k$ joining $x_i$ to $y_i$.
$G$ is $(2k-1)$-connected if size of $G$ is greater than $2k-1$ and $G$ has no separator of size less than $2k-1$.
I have tried proof by contradiction but seem to be stuck, I presume it is fairly easy and am getting frustrated that I cant figure it out??
graph-theory graph-connectivity
graph-theory graph-connectivity
edited Nov 23 at 14:51
hardmath
28.6k94994
asked Nov 23 at 14:23
cxh
1
edited Nov 23 at 14:51
hardmath
28.6k94994
asked Nov 23 at 14:23
cxh
1
edited Nov 23 at 14:51
hardmath
28.6k94994
edited Nov 23 at 14:51
hardmath
28.6k94994
edited Nov 23 at 14:51
hardmath
28.6k94994
28.6k94994
asked Nov 23 at 14:23
cxh
1
asked Nov 23 at 14:23
cxh
1
asked Nov 23 at 14:23
cxh
1
1
1
Please use Latex/MathJax. Could you elaborate on your attempts so far?
– Stockfish
Nov 23 at 14:29
It seems reasonable to leave the title in plain ASCII here. I've added MathJax markup and $LaTeX$ syntax to the body. See this introduction to posting mathematical notation.
– hardmath
Nov 23 at 14:53
add a comment |
1
Please use Latex/MathJax. Could you elaborate on your attempts so far?
– Stockfish
Nov 23 at 14:29
It seems reasonable to leave the title in plain ASCII here. I've added MathJax markup and $LaTeX$ syntax to the body. See this introduction to posting mathematical notation.
– hardmath
Nov 23 at 14:53
1
1
Please use Latex/MathJax. Could you elaborate on your attempts so far?
– Stockfish
Nov 23 at 14:29
Please use Latex/MathJax. Could you elaborate on your attempts so far?
– Stockfish
Nov 23 at 14:29
It seems reasonable to leave the title in plain ASCII here. I've added MathJax markup and $LaTeX$ syntax to the body. See this introduction to posting mathematical notation.
– hardmath
Nov 23 at 14:53
It seems reasonable to leave the title in plain ASCII here. I've added MathJax markup and $LaTeX$ syntax to the body. See this introduction to posting mathematical notation.
– hardmath
Nov 23 at 14:53
add a comment |
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Please use Latex/MathJax. Could you elaborate on your attempts so far?
– Stockfish
Nov 23 at 14:29
It seems reasonable to leave the title in plain ASCII here. I've added MathJax markup and $LaTeX$ syntax to the body. See this introduction to posting mathematical notation.
– hardmath
Nov 23 at 14:53