graph theory: the connectivity is equal to the minimum degree

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Show that if $G$ is simple and $delta geq v-2$,where $delta$ is the minimum degree, then $kappa = delta$, where $kappa$ is the connectivity.






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  • Hi! What is $v$ in relation to $G$? Are there any theorems and techniques you know about that you think might help here?
    – Theo Bendit
    Aug 30 at 4:00










  • @TheoBendit $v$ is the number of vertices
    – Xiaowei Tang
    Aug 30 at 4:01










  • Welcome to Math Stack Exchange! We welcome questions from people of all levels, but strongly encourage people to describe their work and where they got stuck, especially for problems that resemble homework. Please edit your question to include what you tried and why it didn't work.
    – Stella Biderman
    Aug 30 at 4:23














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Show that if $G$ is simple and $delta geq v-2$,where $delta$ is the minimum degree, then $kappa = delta$, where $kappa$ is the connectivity.






graph-theory







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  • Hi! What is $v$ in relation to $G$? Are there any theorems and techniques you know about that you think might help here?
    – Theo Bendit
    Aug 30 at 4:00










  • @TheoBendit $v$ is the number of vertices
    – Xiaowei Tang
    Aug 30 at 4:01










  • Welcome to Math Stack Exchange! We welcome questions from people of all levels, but strongly encourage people to describe their work and where they got stuck, especially for problems that resemble homework. Please edit your question to include what you tried and why it didn't work.
    – Stella Biderman
    Aug 30 at 4:23












up vote
2
down vote

favorite









up vote
2
down vote

favorite











Show that if $G$ is simple and $delta geq v-2$,where $delta$ is the minimum degree, then $kappa = delta$, where $kappa$ is the connectivity.






graph-theory







share|cite|improve this question













Show that if $G$ is simple and $delta geq v-2$,where $delta$ is the minimum degree, then $kappa = delta$, where $kappa$ is the connectivity.






graph-theory




graph-theory






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share|cite|improve this question











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asked Aug 30 at 3:51









Xiaowei Tang

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  • Hi! What is $v$ in relation to $G$? Are there any theorems and techniques you know about that you think might help here?
    – Theo Bendit
    Aug 30 at 4:00










  • @TheoBendit $v$ is the number of vertices
    – Xiaowei Tang
    Aug 30 at 4:01










  • Welcome to Math Stack Exchange! We welcome questions from people of all levels, but strongly encourage people to describe their work and where they got stuck, especially for problems that resemble homework. Please edit your question to include what you tried and why it didn't work.
    – Stella Biderman
    Aug 30 at 4:23
















  • Hi! What is $v$ in relation to $G$? Are there any theorems and techniques you know about that you think might help here?
    – Theo Bendit
    Aug 30 at 4:00










  • @TheoBendit $v$ is the number of vertices
    – Xiaowei Tang
    Aug 30 at 4:01










  • Welcome to Math Stack Exchange! We welcome questions from people of all levels, but strongly encourage people to describe their work and where they got stuck, especially for problems that resemble homework. Please edit your question to include what you tried and why it didn't work.
    – Stella Biderman
    Aug 30 at 4:23















Hi! What is $v$ in relation to $G$? Are there any theorems and techniques you know about that you think might help here?
– Theo Bendit
Aug 30 at 4:00




Hi! What is $v$ in relation to $G$? Are there any theorems and techniques you know about that you think might help here?
– Theo Bendit
Aug 30 at 4:00












@TheoBendit $v$ is the number of vertices
– Xiaowei Tang
Aug 30 at 4:01




@TheoBendit $v$ is the number of vertices
– Xiaowei Tang
Aug 30 at 4:01












Welcome to Math Stack Exchange! We welcome questions from people of all levels, but strongly encourage people to describe their work and where they got stuck, especially for problems that resemble homework. Please edit your question to include what you tried and why it didn't work.
– Stella Biderman
Aug 30 at 4:23




Welcome to Math Stack Exchange! We welcome questions from people of all levels, but strongly encourage people to describe their work and where they got stuck, especially for problems that resemble homework. Please edit your question to include what you tried and why it didn't work.
– Stella Biderman
Aug 30 at 4:23










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The minimum degree of a graph can never be larger than $v$. So, you have that $vgeqdeltageq v-2$. However, we can even strengthen this inequality further. Do you see how?



Due to how strong this inequality is, we can come up with a very strong characterization of what graphs fit that inequality. This characterization should look like "$G$ is either a ____ graph or a _____ graph with _______." This characterization should make it clear that $kappa =delta$. The first two blanks are both filled in with the same word, while the last is a statement about the structure of $G$.






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    up vote
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    down vote













    The minimum degree of a graph can never be larger than $v$. So, you have that $vgeqdeltageq v-2$. However, we can even strengthen this inequality further. Do you see how?



    Due to how strong this inequality is, we can come up with a very strong characterization of what graphs fit that inequality. This characterization should look like "$G$ is either a ____ graph or a _____ graph with _______." This characterization should make it clear that $kappa =delta$. The first two blanks are both filled in with the same word, while the last is a statement about the structure of $G$.






    share|cite|improve this answer
























      up vote
      1
      down vote













      The minimum degree of a graph can never be larger than $v$. So, you have that $vgeqdeltageq v-2$. However, we can even strengthen this inequality further. Do you see how?



      Due to how strong this inequality is, we can come up with a very strong characterization of what graphs fit that inequality. This characterization should look like "$G$ is either a ____ graph or a _____ graph with _______." This characterization should make it clear that $kappa =delta$. The first two blanks are both filled in with the same word, while the last is a statement about the structure of $G$.






      share|cite|improve this answer






















        up vote
        1
        down vote










        up vote
        1
        down vote









        The minimum degree of a graph can never be larger than $v$. So, you have that $vgeqdeltageq v-2$. However, we can even strengthen this inequality further. Do you see how?



        Due to how strong this inequality is, we can come up with a very strong characterization of what graphs fit that inequality. This characterization should look like "$G$ is either a ____ graph or a _____ graph with _______." This characterization should make it clear that $kappa =delta$. The first two blanks are both filled in with the same word, while the last is a statement about the structure of $G$.






        share|cite|improve this answer












        The minimum degree of a graph can never be larger than $v$. So, you have that $vgeqdeltageq v-2$. However, we can even strengthen this inequality further. Do you see how?



        Due to how strong this inequality is, we can come up with a very strong characterization of what graphs fit that inequality. This characterization should look like "$G$ is either a ____ graph or a _____ graph with _______." This characterization should make it clear that $kappa =delta$. The first two blanks are both filled in with the same word, while the last is a statement about the structure of $G$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 30 at 4:20









        Stella Biderman

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