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    \begin{center}
    \Title{The almost majority neighbor sum distinguishing edge coloring}

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    \Author{El\.zbieta Sidorowicz and \underline{El\.zbieta Turowska}}
     {University of Zielona G\'ora}
    \Email{e.sidorowicz@im.uz.zgora.pl, e.turowska@im.uz.zgora.pl}
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 \begin{abstract}
A  $k$-edge coloring  of a graph with colors in $[k]$ is  neighbor sum distinguishing if, for any two adjacent vertices, the sums of the colors of the edges incident with each of them are distinct. The smallest value of $k$ such that a neighbor sum distinguishing $k$-coloring of $G$ exists is denoted by $\chi^{e}_{\sum}(G)$. When we add the additional restriction that the edge $k$-coloring must be proper, then   the smallest value of $k$ such that such a coloring exists is denoted by $\chi'_{\sum}(G)$. The first type of  coloring is related with the 1-2-3 Conjecture, which was already proven by Keusch (\cite{Ke24}). The second type of coloring is related with another conjecture proposed by Flandrin et al. (\cite{FlMa13}), which states   that $\chi'_{\sum}(G)\le \Delta(G)+2$ for any  graph $G$ with  no components isomorphic to $K_2$ and $G \neq C_5$. This conjecture remains open. 

We consider an edge coloring that is on one hand stronger than the edge coloring in the 1-2-3 Conjecture, and on the other hand weaker than the coloring in conjecture proposed by Flandrin et al.. An~edge $k$-coloring of a graph $G$~is called almost majority   if for every vertex $v \in V(G)$ and every color $\alpha\in [k]$ at most $\left\lceil d(v)/2\right\rceil$ edges incident to $v$~have the color~$\alpha$. 
An edge $k$-coloring of a graph $G$~is called almost majority neighbor sum distinguishing  if it is almost majority and neighbor sum distinguishing. The minimum value of $k$~for which there exists such an edge $k$-coloring of a graph $G$ is called the almost majority neighbor sum distinguishing index of a graph~$G$~and is denoted by $\chi_{\sum}^{AM}(G)$. We study  $\chi_{\sum}^{AM}(G)$ for some   classes of graphs.
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    \begin{thebibliography}{99}
    
    \bibitem{FlMa13} E. Flandrin, A. Marczyk, J. Przyby{\l}o, J-F. Sacle, M. Wo\'zniak,  Neighbour sum distinguishing index. Graphs Combin. 29(5) (2013), 1329--1336.
    
    \bibitem{Ke24} R. Keusch, A Solution to the 1-2-3 Conjecture. J. Combin. Theory Ser. B 166 (2024) 182--202.
    \end{thebibliography}





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