A note on adjacent vertex distinguishing edge coloring of graphs

An adjacent vertex distinguishing edge coloring of graphs is a proper edge coloring of graphs such that no two adjacent vertices have the same set of colors, and the minimum number of the colors required for such coloring is called adjacent vertex distinguishing edge chromatic number. In this paper, we characterize the relationships of adjacent vertex distinguishing edge chromatic numbers between all bicyclic graphs as well as some tricyclic base graphs and their subgraphs, respectively. By the way, the open problem proposed by Zhang et al. [Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15(5) (2002) 623–626] is answered completely on a bicyclic graph.