Vertex distinguishing colorings of graphs with Δ (G)=2

In a paper by Burris and Schelp (J. Graph Theory 26 (2) (1997) 70), a conjecture was made concerning the number of colors chi'(s)(G) required to proper edge-color G so that each vertex has a distinct set of colors incident to it. We consider the case when Delta(G) = 2, so that G is a union of paths and cycles. In particular we find the exact values of chi'(s)(G) and hence verify the conjecture when G consists of just paths or just cycles. We also give good bounds on chi'(s)(G) s when G contains both paths and cycles. (C) 2002 Elsevier Science B.V. All rights reserved.