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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.

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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 vertex colouring c of a graph G is called nonrepetitive if for every integer r ≥ 1 and every path P = (v 1 , v 2 , . . . , v 2r ) in G, the first half of P is coloured differently from the second half of P, that is, c(v j ) = c(v r+j ) for some j = 1, 2, . . . , r. This notion was inspired by a striking result of Thue asserting that the path P n on n vertices has a nonrepetitive three-colouring, no matter how large n is. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The Thue choice number of G, denoted by π ch (G), is the least integer k such that for every k-list assignment L there is a nonrepetitive colouring c of G satisfying c(v) ∈ L(v) for every vertex v of G. Using the Lefthanded Local Lemma we prove that π ch (P n ) ≤ 4 for every n.