r-Strong edge colorings of graphs Original Research Article

Let G be a graph and for any natural number r, image denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623–626] it has been proved that for any tree T with at least three vertices, image. Here we generalize this result and show that image. Moreover, we show that if for any two vertices image and image with maximum degree image, then image. Also for any tree T with image we prove that image. Finally, it is shown that for any graph G with no isolated edges, image.