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

Suppose that for any bipartite planar graph $$G=(V,E)$$, we can find a vertex labeling $$c:V\to \{1,2,3\}$$ such that for any two adjacent vertices $$u$$ and $$w$$: $$c(u)-\sum_{v\in N(u)}c(v)\neq c(w)-\sum_{v\in N(w)}c(v)$$ where $$N(v)$$ denotes the neighborhood of the vertex $$v\in V$$. 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: $$\sum_{v\in N(u)}c(v)\neq \sum_{v\in N(w)}c(v)?$$

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 $$\sum_{v\in N(u)}c(v)=\sum_{v\in N(w)}c(v)$$ holds then $$c(u)\neq c(w)$$. Without loss of generality, we may assume that $$c(u). Then, change the label of $$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").