Definitions of $epsilon$-regular partition












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I am wondering about the equivalence between two definitions of an $epsilon$-regular partition of a graph.



First of all, if $G$ is a graph and $A$ and $B$ are subsets of its vertex set, the density of edges between $A$ and $B$ is defined as
$$d(A,B)=frac{|E(A,B)|}{|A||B|},$$
where $|E(A,B)|$ is the number of edges between $A$ and $B$.



Given some $epsilon>0$, the pair $(A,B)$ is said to be $epsilon$-regular if, for every $A'subseteq A$ and $B'subseteq B$ with $|A'|geq epsilon |A|$ and $|B'|geq epsilon |B|$, we have that
$$|d(A',B')-d(A,B)|leq epsilon.$$



Now, what we want to do next is to define what it means for a partition of the vertex set to be $epsilon$-regular, and I have found two different definitions (from different sources).




Definition 1. Given a graph $G$ on $n$ vertices and an $epsilon>0$, a partition ${X_1, dots, X_k}$ of its vertex set is $epsilon$-regular if
$$sum frac{|X_i||X_j|}{n^2} leq epsilon,$$
where the sum is taken over all pairs $(X_i,X_j)$ which are not $epsilon$-regular.



Definition 2. Given a graph $G$ on $n$ vertices and an $epsilon>0$, a partition ${V_0, V_1, dots, V_k}$ of its vertex set $V$ is $epsilon$-regular if:




  • $|V_0|leq epsilon |V|$

  • $|V_1| = dots = |V_k|$

  • at most $epsilonbinom{k}{2}$ pairs $(V_i,V_j)$ are not $epsilon$-regular.




I am assuming that these definitions are equivalent, and they are both used in various different sources in order to prove Szemeredi's regularity lemma, but I cannot see if that is indeed true.



Can anyone help?



Any help is much appreciated.






graph-theory definition







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    0












    $begingroup$


    I am wondering about the equivalence between two definitions of an $epsilon$-regular partition of a graph.



    First of all, if $G$ is a graph and $A$ and $B$ are subsets of its vertex set, the density of edges between $A$ and $B$ is defined as
    $$d(A,B)=frac{|E(A,B)|}{|A||B|},$$
    where $|E(A,B)|$ is the number of edges between $A$ and $B$.



    Given some $epsilon>0$, the pair $(A,B)$ is said to be $epsilon$-regular if, for every $A'subseteq A$ and $B'subseteq B$ with $|A'|geq epsilon |A|$ and $|B'|geq epsilon |B|$, we have that
    $$|d(A',B')-d(A,B)|leq epsilon.$$



    Now, what we want to do next is to define what it means for a partition of the vertex set to be $epsilon$-regular, and I have found two different definitions (from different sources).




    Definition 1. Given a graph $G$ on $n$ vertices and an $epsilon>0$, a partition ${X_1, dots, X_k}$ of its vertex set is $epsilon$-regular if
    $$sum frac{|X_i||X_j|}{n^2} leq epsilon,$$
    where the sum is taken over all pairs $(X_i,X_j)$ which are not $epsilon$-regular.



    Definition 2. Given a graph $G$ on $n$ vertices and an $epsilon>0$, a partition ${V_0, V_1, dots, V_k}$ of its vertex set $V$ is $epsilon$-regular if:




    • $|V_0|leq epsilon |V|$

    • $|V_1| = dots = |V_k|$

    • at most $epsilonbinom{k}{2}$ pairs $(V_i,V_j)$ are not $epsilon$-regular.




    I am assuming that these definitions are equivalent, and they are both used in various different sources in order to prove Szemeredi's regularity lemma, but I cannot see if that is indeed true.



    Can anyone help?



    Any help is much appreciated.






    graph-theory definition







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I am wondering about the equivalence between two definitions of an $epsilon$-regular partition of a graph.



      First of all, if $G$ is a graph and $A$ and $B$ are subsets of its vertex set, the density of edges between $A$ and $B$ is defined as
      $$d(A,B)=frac{|E(A,B)|}{|A||B|},$$
      where $|E(A,B)|$ is the number of edges between $A$ and $B$.



      Given some $epsilon>0$, the pair $(A,B)$ is said to be $epsilon$-regular if, for every $A'subseteq A$ and $B'subseteq B$ with $|A'|geq epsilon |A|$ and $|B'|geq epsilon |B|$, we have that
      $$|d(A',B')-d(A,B)|leq epsilon.$$



      Now, what we want to do next is to define what it means for a partition of the vertex set to be $epsilon$-regular, and I have found two different definitions (from different sources).




      Definition 1. Given a graph $G$ on $n$ vertices and an $epsilon>0$, a partition ${X_1, dots, X_k}$ of its vertex set is $epsilon$-regular if
      $$sum frac{|X_i||X_j|}{n^2} leq epsilon,$$
      where the sum is taken over all pairs $(X_i,X_j)$ which are not $epsilon$-regular.



      Definition 2. Given a graph $G$ on $n$ vertices and an $epsilon>0$, a partition ${V_0, V_1, dots, V_k}$ of its vertex set $V$ is $epsilon$-regular if:




      • $|V_0|leq epsilon |V|$

      • $|V_1| = dots = |V_k|$

      • at most $epsilonbinom{k}{2}$ pairs $(V_i,V_j)$ are not $epsilon$-regular.




      I am assuming that these definitions are equivalent, and they are both used in various different sources in order to prove Szemeredi's regularity lemma, but I cannot see if that is indeed true.



      Can anyone help?



      Any help is much appreciated.






      graph-theory definition







      share|cite|improve this question









      $endgroup$




      I am wondering about the equivalence between two definitions of an $epsilon$-regular partition of a graph.



      First of all, if $G$ is a graph and $A$ and $B$ are subsets of its vertex set, the density of edges between $A$ and $B$ is defined as
      $$d(A,B)=frac{|E(A,B)|}{|A||B|},$$
      where $|E(A,B)|$ is the number of edges between $A$ and $B$.



      Given some $epsilon>0$, the pair $(A,B)$ is said to be $epsilon$-regular if, for every $A'subseteq A$ and $B'subseteq B$ with $|A'|geq epsilon |A|$ and $|B'|geq epsilon |B|$, we have that
      $$|d(A',B')-d(A,B)|leq epsilon.$$



      Now, what we want to do next is to define what it means for a partition of the vertex set to be $epsilon$-regular, and I have found two different definitions (from different sources).




      Definition 1. Given a graph $G$ on $n$ vertices and an $epsilon>0$, a partition ${X_1, dots, X_k}$ of its vertex set is $epsilon$-regular if
      $$sum frac{|X_i||X_j|}{n^2} leq epsilon,$$
      where the sum is taken over all pairs $(X_i,X_j)$ which are not $epsilon$-regular.



      Definition 2. Given a graph $G$ on $n$ vertices and an $epsilon>0$, a partition ${V_0, V_1, dots, V_k}$ of its vertex set $V$ is $epsilon$-regular if:




      • $|V_0|leq epsilon |V|$

      • $|V_1| = dots = |V_k|$

      • at most $epsilonbinom{k}{2}$ pairs $(V_i,V_j)$ are not $epsilon$-regular.




      I am assuming that these definitions are equivalent, and they are both used in various different sources in order to prove Szemeredi's regularity lemma, but I cannot see if that is indeed true.



      Can anyone help?



      Any help is much appreciated.






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      asked Dec 8 '18 at 15:49









      IlefenIlefen

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          It is easy to check that for each $epsilon>0$ each graph, which is $epsilon$-regular according to Definition 2 is $epsilon$-regular according to Definition 1. But not conversely, because according to Definition 1, any partition of any finite graph is $1$-regular, whereas Definition 2 imposes additional restrictions on the sizes of partition members.






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            It is easy to check that for each $epsilon>0$ each graph, which is $epsilon$-regular according to Definition 2 is $epsilon$-regular according to Definition 1. But not conversely, because according to Definition 1, any partition of any finite graph is $1$-regular, whereas Definition 2 imposes additional restrictions on the sizes of partition members.






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              $begingroup$

              It is easy to check that for each $epsilon>0$ each graph, which is $epsilon$-regular according to Definition 2 is $epsilon$-regular according to Definition 1. But not conversely, because according to Definition 1, any partition of any finite graph is $1$-regular, whereas Definition 2 imposes additional restrictions on the sizes of partition members.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                It is easy to check that for each $epsilon>0$ each graph, which is $epsilon$-regular according to Definition 2 is $epsilon$-regular according to Definition 1. But not conversely, because according to Definition 1, any partition of any finite graph is $1$-regular, whereas Definition 2 imposes additional restrictions on the sizes of partition members.






                share|cite|improve this answer









                $endgroup$



                It is easy to check that for each $epsilon>0$ each graph, which is $epsilon$-regular according to Definition 2 is $epsilon$-regular according to Definition 1. But not conversely, because according to Definition 1, any partition of any finite graph is $1$-regular, whereas Definition 2 imposes additional restrictions on the sizes of partition members.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 9 '18 at 7:12









                Alex RavskyAlex Ravsky

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