Geodesic
An Improved Upper Bound on Neighbor Expanded Sum Distinguishing Index
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 323-329

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A total k-weighting f of a graph G is an assignment of integers from the set 1, . . ., k to the vertices and edges of G. We say that f is neighbor expanded sum distinguishing, or NESD for short, if Σw∈N(v) (f(vw) + f(w)) differs from Σw∈N(u)(f(uw) + f(w)) for every two adjacent vertices v and u of G. The neighbor expanded sum distinguishing index of G, denoted by egndiΣ(G), is the minimum positive integer k for which there exists an NESD weighting of G. An NESD weighting was introduced and investigated by Flandrin et al. (2017), where they conjectured that egndiΣ(G) ≤ 2 for any graph G. They examined some special classes of graphs, while proving that egndiΣ(G) ≤ χ(G) + 1. We improve this bound and show that egndiΣ(G) ≤ 3 for any graph G. We also show that the conjecture holds for all bipartite, 3-regular and 4-regular graphs.
Keywords: general edge coloring, total coloring, neighbor sum distinguishing index 
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Vučković, Bojan. An Improved Upper Bound on Neighbor Expanded Sum Distinguishing Index. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 323-329. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a21/