, Volume 29, Issue 5, pp 1329-1336,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 26 May 2012

Neighbor Sum Distinguishing Index

Abstract

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 ndiΣ(G). In the paper we conjecture that for any connected graph G ≠ C 5 of order n ≥ 3 we have ndiΣ(G) ≤ Δ(G) + 2. We prove this conjecture for several classes of graphs. We also show that ndiΣ(G) ≤ 7Δ(G)/2 for any graph G with Δ(G) ≥ 2 and ndiΣ(G) ≤ 8 if G is cubic.

The work of the authors from AGH University of Science and Technology was partially supported by the Polish Ministry of Science and Higher Education.