We consider proper edge colorings of a graph G using colors of the set {1, . . . , k}. Such a coloring is called neighbor sum distinguishing if for any pair of adjacent vertices x and y the sum of colors taken on the edges incident to x is different from the sum of colors taken on the edges incident to y. The smallest value of k in such a coloring of G is denoted by ndi (G). In the paper we conjecture that for any connected graph G = C 5 of order n ≥ 3 we have ndi (G) ≤ (G) + 2. We prove this conjecture for several classes of graphs. We also show that ndi (G) ≤ 7 (G)/2 for any graph G with (G) ≥ 2 and ndi (G) ≤ 8 if G is cubic.

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a 1 , . . . , a k ) of positive integers such that a 1 +. . .+a k = n there exists a partition (V 1 , . . . , V k ) of the vertex set of G such that for each i ∈ {1, . . . , k}, V i induces a connected subgraph of G on a i vertices.D. Barth and H. Fournier showed that if a tree T is arbitrarily vertex decomposable, then T has maximum degree at most 4. In this paper we give a complete characterization of arbitrarily vertex decomposable caterpillars with four leaves. We also describe two families of 292

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