A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors from \([k]=\{1,2,\cdots,k\}\) . The neighbor set distinguishing index ndi(G) of G is the smallest integer k for which G admits a proper edge k-coloring such that any pair of adjacent vertices are incident to distinct sets of colors. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge \(uv\in E(G)\) , the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By nsdi(G), we denote the smallest value k in such a coloring of G. Apparently, for any graph G, \(ndi(G)\le nsdi(G)\) . Wang and Wang (Applied Mathematics Letters 24 (2011) 2034-2037) proved that if G is a \(K_4\) -minor free graph with \(\Delta (G)\ge 5\) , then \(ndi(G)\le \Delta (G)+1\) . They posed an open problem: for a \(K_4\) -minor free graph G with \(\Delta (G)\in \{3,4\}\) , is it ture that \(ndi(G)\le \Delta (G)+1\) ? In addition, Zhang, Ding, Wang, Yan and Zhou (Graphs and combinatorics 32 (2016) 1621-1633) showed that if G is a \(K_4\) -minor free graph with \(\Delta (G)\ge 5\) , then \(nsdi(G)\le \Delta (G)+1\) . They also proposed an open problem: let G be a \(K_4\) -minor free graph with \(\Delta (G)\in \{3,4\}\) . Does it holds that \(nsdi(G)\le \Delta (G)+1\) ? In this paper, we show that the problems above with \(\Delta (G)=4\) are true, improving a known result of Wang and Wang.
