Quasi-majority neighbor sum distinguishing edge-colorings

In this paper, a $k$-edge-coloring of $G$ is any mapping $c:E(G)\longrightarrow [k]$. The edge-coloring $c$ of $G$ naturally defines a vertex-coloring $\sigma_{c}: V(G) \to \mathbb{N}$, where $\sigma_{c}(v)=\sum_{u\in N_G(v)}c(vu)$ for every vertex $v\in V(G)$. The edge-coloring $c$ is said to be neighbor sum distinguishing if it results in a proper vertex-coloring $\sigma_{c}$, which that $\sigma_{c}(u) \neq \sigma_{c}(v)$ for every edge $uv$ in $G$. We investigate neighbor sum distinguishing edge-colorings with local constraints, where the edge-coloring is quasi-majority at each vertex. Specifically, every vertex $v$ is incident to at most $\left\lceil d(v)/2 \right\rceil$ edges of one color. This type of coloring is referred to as quasi-majority neighbor sum distinguishing edge-coloring. The minimum number of colors required for a graph to have a quasi-majority neighbor sum distinguishing edge-coloring is called the quasi-majority neighbor sum distinguishing index. A graph is nice if it has no component isomorphic to $K_2$. We prove that any nice graph admits a quasi-majority neighbor sum distinguishing edge-coloring using at most 12 colors. This bound can be improved for bipartite graphs and graphs with a maximum degree of at most 4. Specifically, we show that every nice bipartite graph can be colored with 6 colors, and every nice graph with a maximum degree of at most 4 can be colored with 7 colors. Additionally, we determine the exact value of the quasi-majority neighbor sum distinguishing index for complete graphs, complete bipartite graphs, and trees. We also consider majority neighbor sum distinguishing edge-colorings, that is, when each vertex is incident to at most $d(v)/2$ edges with the same color.