Prove that the property is preserved under isomorphism.












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Prove that the following properties: ‘There exists a vertex of degree 4’ and ‘There exists a cycle of length 4’ are preserved under isomorphism.




By definition, a property of a graph is said to be preserved under
isomorphism if whenever G has that property, every graph isomorphic to G also
has that property.



I know how reason about the first property, for example: since an isomorphism is a bijection between sets of vertices, isomorphic graphs must have the same number of vertices. If one graph has a vertex of degree 4, then another graph should also have a vertex of the same degree so they can be isomorphic. Thus, $v$ and $f(v)$ will have the same degree.



Would it be sufficient to say this for the first? I am not sure how to construct a formal proof for these properties. How can one prove the second property?










share|cite|improve this question









$endgroup$

















    1












    $begingroup$



    Prove that the following properties: ‘There exists a vertex of degree 4’ and ‘There exists a cycle of length 4’ are preserved under isomorphism.




    By definition, a property of a graph is said to be preserved under
    isomorphism if whenever G has that property, every graph isomorphic to G also
    has that property.



    I know how reason about the first property, for example: since an isomorphism is a bijection between sets of vertices, isomorphic graphs must have the same number of vertices. If one graph has a vertex of degree 4, then another graph should also have a vertex of the same degree so they can be isomorphic. Thus, $v$ and $f(v)$ will have the same degree.



    Would it be sufficient to say this for the first? I am not sure how to construct a formal proof for these properties. How can one prove the second property?










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$



      Prove that the following properties: ‘There exists a vertex of degree 4’ and ‘There exists a cycle of length 4’ are preserved under isomorphism.




      By definition, a property of a graph is said to be preserved under
      isomorphism if whenever G has that property, every graph isomorphic to G also
      has that property.



      I know how reason about the first property, for example: since an isomorphism is a bijection between sets of vertices, isomorphic graphs must have the same number of vertices. If one graph has a vertex of degree 4, then another graph should also have a vertex of the same degree so they can be isomorphic. Thus, $v$ and $f(v)$ will have the same degree.



      Would it be sufficient to say this for the first? I am not sure how to construct a formal proof for these properties. How can one prove the second property?










      share|cite|improve this question









      $endgroup$





      Prove that the following properties: ‘There exists a vertex of degree 4’ and ‘There exists a cycle of length 4’ are preserved under isomorphism.




      By definition, a property of a graph is said to be preserved under
      isomorphism if whenever G has that property, every graph isomorphic to G also
      has that property.



      I know how reason about the first property, for example: since an isomorphism is a bijection between sets of vertices, isomorphic graphs must have the same number of vertices. If one graph has a vertex of degree 4, then another graph should also have a vertex of the same degree so they can be isomorphic. Thus, $v$ and $f(v)$ will have the same degree.



      Would it be sufficient to say this for the first? I am not sure how to construct a formal proof for these properties. How can one prove the second property?







      discrete-mathematics graph-theory






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      asked Dec 4 '18 at 22:34









      JohaJoha

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          $begingroup$

          A graph isomorphism is a bijection, but it also preserves the edge relations between vertices. This is the main point of the exercise.



          To illustrate, consider the degree $4$ vertex $u$. Since it has degree $4$, there are four vertices attached to it by edges, say those vertices are $v_1,v_2,v_3,v_4$. Then because $uv_1$ is an edge, and $f$ is an isomorphism, then $f(uv_1)$ is an edge between $f(u)$ and $f(v_1)$.



          This argument is not complete yet. How do we know $f(u)$ is degree $4$ exactly? That is, it doesn't have more/less adjacent vertices than we expect.






          share|cite|improve this answer









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            $begingroup$

            A graph isomorphism is a bijection, but it also preserves the edge relations between vertices. This is the main point of the exercise.



            To illustrate, consider the degree $4$ vertex $u$. Since it has degree $4$, there are four vertices attached to it by edges, say those vertices are $v_1,v_2,v_3,v_4$. Then because $uv_1$ is an edge, and $f$ is an isomorphism, then $f(uv_1)$ is an edge between $f(u)$ and $f(v_1)$.



            This argument is not complete yet. How do we know $f(u)$ is degree $4$ exactly? That is, it doesn't have more/less adjacent vertices than we expect.






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              A graph isomorphism is a bijection, but it also preserves the edge relations between vertices. This is the main point of the exercise.



              To illustrate, consider the degree $4$ vertex $u$. Since it has degree $4$, there are four vertices attached to it by edges, say those vertices are $v_1,v_2,v_3,v_4$. Then because $uv_1$ is an edge, and $f$ is an isomorphism, then $f(uv_1)$ is an edge between $f(u)$ and $f(v_1)$.



              This argument is not complete yet. How do we know $f(u)$ is degree $4$ exactly? That is, it doesn't have more/less adjacent vertices than we expect.






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                A graph isomorphism is a bijection, but it also preserves the edge relations between vertices. This is the main point of the exercise.



                To illustrate, consider the degree $4$ vertex $u$. Since it has degree $4$, there are four vertices attached to it by edges, say those vertices are $v_1,v_2,v_3,v_4$. Then because $uv_1$ is an edge, and $f$ is an isomorphism, then $f(uv_1)$ is an edge between $f(u)$ and $f(v_1)$.



                This argument is not complete yet. How do we know $f(u)$ is degree $4$ exactly? That is, it doesn't have more/less adjacent vertices than we expect.






                share|cite|improve this answer









                $endgroup$



                A graph isomorphism is a bijection, but it also preserves the edge relations between vertices. This is the main point of the exercise.



                To illustrate, consider the degree $4$ vertex $u$. Since it has degree $4$, there are four vertices attached to it by edges, say those vertices are $v_1,v_2,v_3,v_4$. Then because $uv_1$ is an edge, and $f$ is an isomorphism, then $f(uv_1)$ is an edge between $f(u)$ and $f(v_1)$.



                This argument is not complete yet. How do we know $f(u)$ is degree $4$ exactly? That is, it doesn't have more/less adjacent vertices than we expect.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 4 '18 at 22:59









                Larry B.Larry B.

                2,801828




                2,801828






























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