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A weighting of the edges of a graph is called irregular if the weighted degrees of the vertices are all different. In this note we show that such a weighting is possible from the weight set {1, 2, . . . , 6 n δ } for all graphs not containing a component with exactly 2 vertices or two isolated vertices.

Consider a simple graph G. A labeling w :The goal is to obtain a total vertex product-irregular labeling that minimizes the maximum label. This minimum value is called the total vertex product irregularity strength and denoted tvps(G). In this paper we provide some general lower and upper bounds, as well as exact values for chosen families of graphs.

Abstract. A weighting of the edges of a hypergraph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this paper we show that such a weighting is possible from the weight set {1, 2, . . . , r + 1} for all linear hypergraphs with maximum edge size r ≥ 4 and not containing isolated edges. The number r + 1 is best possible for this statement.Further, the weight set {1, 2, 3, 4, 5} is sufficient for all hypergraphs with maximum edge size 3, as well as {1, 2, . . . , 5r − 5} for all hypergraphs with maximum edge size r ≥ 4, up to some trivial exceptions.