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A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ 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 ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ max {2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ max {2Δ(G), 19} if G is a graph such that mad(G) ≤ 5.

A graph is said to be equitably k-colorable if the vertex set V (G) can be partitioned into k independent subsets V 1 , V 2 ,. .. , V k such that ||V i | − |V j || ≤ 1 (1 ≤ i, j ≤ k). A graph G is equitably k-choosable if, for any given k-uniform list assignment L, G is L-colorable and each color appears on at most |V (G)| k vertices. In this paper, we prove that if G is a graph such that mad(G) < 3, then G is equitably k-colorable and equitably kchoosable where k ≥ max{∆(G), 4}. Moreover, if G is a graph such that mad(G) < 12 5 , then G is equitably k-colorable and equitably k-choosable where k ≥ max{∆(G), 3}.

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A proper [k]-total coloring c of a graph G is a mapping c from V (G) E(G) to [k] = {1, 2, • • • , k} such that c(x) = c(y) for which x, y ∈ V (G) E(G) and x is adjacent to or incident with y. Let (v) denote the product of c(v) and the colors on all the edges incident with v. For each edge uv ∈ E(G), if (u) = (v), then the coloring c is called a neighbor product distinguishing total coloring of G. we use χ (G) to denote the minimal value of k in such a coloring of G. In 2015, Li et al. conjectured that ∆(G) + 3 colors enable a graph to have a neighbor product distinguishing total coloring. In this paper, we consider the neighbor product distinguishing total coloring of corona product G • H, and obtain that χ (G • H) ≤ ∆(G • H) + 3.