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An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices admits different sets of colors. The minimum number of colors needed for such a coloring of G is denoted by χ'a(G). In this paper, we show that if G is a planar graph with maximum degree Δ ≥ 16, then Δ ≤ χ'a(G) ≤ Δ + 1, and χ'a(G) = Δ+1 if and only if G contains two adjacent vertices of maximum degree. [ABSTRACT FROM AUTHOR]

It is known that the bandwidth sum of a hypergraph, over all the labellings of the vertices with distinct integers, is the minimum in the total sum of the maximum difference between the labels of the vertices in one edge. Let H be a hypergraph. Denote bandwidth sum of H by BS(H) and the bandwidth sum of a graph G by BS(G). In this paper, the relationship between BS(H) and BS(G) is given. [ABSTRACT FROM AUTHOR]