Question on Vertex Labeling (Related to Lucky Labeling of Graphs)












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Suppose that for any bipartite planar graph $G=(V,E)$, we can find a vertex labeling $c:Vto {1,2,3}$ such that for any two adjacent vertices $u$ and $w$:
$$c(u)-sum_{vin N(u)}c(v)neq c(w)-sum_{vin N(w)}c(v)$$ where $N(v)$ denotes the neighborhood of the vertex $vin V$. My question is: is it true that there also must exist a labeling of $G$ with labels ${1,2,3}$ such that for any two adjacent vertices $u$ and $w$, we have:
$$sum_{vin N(u)}c(v)neq sum_{vin N(w)}c(v)?$$



It is possible for two adjacent vertices to satisfy the first equation, but not the second in some labeling $c$. My idea was to modify the initial labeling in a way that makes the second inequality hold. That is, for some adjacent vertices $u$ and $w$ satisfying the first equation, if $sum_{vin N(u)}c(v)=sum_{vin N(w)}c(v)$ holds then $c(u)neq c(w)$. Without loss of generality, we may assume that $c(u)<c(w)$. Then, change the label of $u$ to $c(w)$ thus obtaining the new labeling $c'$. However, this may affect the relationship of $u$ with its other neighbors and my attempts to analyze those weren't successful. I would really appreciate some help.



This question comes from reading the paper of Lason where he seems to claim that the second result is a consequence of the first (if I understand correctly what he means by the "special case").










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    2












    $begingroup$


    Suppose that for any bipartite planar graph $G=(V,E)$, we can find a vertex labeling $c:Vto {1,2,3}$ such that for any two adjacent vertices $u$ and $w$:
    $$c(u)-sum_{vin N(u)}c(v)neq c(w)-sum_{vin N(w)}c(v)$$ where $N(v)$ denotes the neighborhood of the vertex $vin V$. My question is: is it true that there also must exist a labeling of $G$ with labels ${1,2,3}$ such that for any two adjacent vertices $u$ and $w$, we have:
    $$sum_{vin N(u)}c(v)neq sum_{vin N(w)}c(v)?$$



    It is possible for two adjacent vertices to satisfy the first equation, but not the second in some labeling $c$. My idea was to modify the initial labeling in a way that makes the second inequality hold. That is, for some adjacent vertices $u$ and $w$ satisfying the first equation, if $sum_{vin N(u)}c(v)=sum_{vin N(w)}c(v)$ holds then $c(u)neq c(w)$. Without loss of generality, we may assume that $c(u)<c(w)$. Then, change the label of $u$ to $c(w)$ thus obtaining the new labeling $c'$. However, this may affect the relationship of $u$ with its other neighbors and my attempts to analyze those weren't successful. I would really appreciate some help.



    This question comes from reading the paper of Lason where he seems to claim that the second result is a consequence of the first (if I understand correctly what he means by the "special case").










    share|cite|improve this question











    $endgroup$















      2












      2








      2


      4



      $begingroup$


      Suppose that for any bipartite planar graph $G=(V,E)$, we can find a vertex labeling $c:Vto {1,2,3}$ such that for any two adjacent vertices $u$ and $w$:
      $$c(u)-sum_{vin N(u)}c(v)neq c(w)-sum_{vin N(w)}c(v)$$ where $N(v)$ denotes the neighborhood of the vertex $vin V$. My question is: is it true that there also must exist a labeling of $G$ with labels ${1,2,3}$ such that for any two adjacent vertices $u$ and $w$, we have:
      $$sum_{vin N(u)}c(v)neq sum_{vin N(w)}c(v)?$$



      It is possible for two adjacent vertices to satisfy the first equation, but not the second in some labeling $c$. My idea was to modify the initial labeling in a way that makes the second inequality hold. That is, for some adjacent vertices $u$ and $w$ satisfying the first equation, if $sum_{vin N(u)}c(v)=sum_{vin N(w)}c(v)$ holds then $c(u)neq c(w)$. Without loss of generality, we may assume that $c(u)<c(w)$. Then, change the label of $u$ to $c(w)$ thus obtaining the new labeling $c'$. However, this may affect the relationship of $u$ with its other neighbors and my attempts to analyze those weren't successful. I would really appreciate some help.



      This question comes from reading the paper of Lason where he seems to claim that the second result is a consequence of the first (if I understand correctly what he means by the "special case").










      share|cite|improve this question











      $endgroup$




      Suppose that for any bipartite planar graph $G=(V,E)$, we can find a vertex labeling $c:Vto {1,2,3}$ such that for any two adjacent vertices $u$ and $w$:
      $$c(u)-sum_{vin N(u)}c(v)neq c(w)-sum_{vin N(w)}c(v)$$ where $N(v)$ denotes the neighborhood of the vertex $vin V$. My question is: is it true that there also must exist a labeling of $G$ with labels ${1,2,3}$ such that for any two adjacent vertices $u$ and $w$, we have:
      $$sum_{vin N(u)}c(v)neq sum_{vin N(w)}c(v)?$$



      It is possible for two adjacent vertices to satisfy the first equation, but not the second in some labeling $c$. My idea was to modify the initial labeling in a way that makes the second inequality hold. That is, for some adjacent vertices $u$ and $w$ satisfying the first equation, if $sum_{vin N(u)}c(v)=sum_{vin N(w)}c(v)$ holds then $c(u)neq c(w)$. Without loss of generality, we may assume that $c(u)<c(w)$. Then, change the label of $u$ to $c(w)$ thus obtaining the new labeling $c'$. However, this may affect the relationship of $u$ with its other neighbors and my attempts to analyze those weren't successful. I would really appreciate some help.



      This question comes from reading the paper of Lason where he seems to claim that the second result is a consequence of the first (if I understand correctly what he means by the "special case").







      combinatorics graph-theory algebraic-combinatorics






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      share|cite|improve this question













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      edited Dec 10 '18 at 6:36







      Yulia Alexandr

















      asked Dec 10 '18 at 5:55









      Yulia AlexandrYulia Alexandr

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