Graceful labeling is assignment of distinct labels from f {0,…., |E|} to vertices of graph, where edges are labeled by difference of absolute values of adjacent vertices and every label from f {1,2,….,|E|} is used exactly once as an edge label. The edge gracefulness is reversal of gracefulness. It involves labeling edges, rather than vertices, with an induced numbering consisting of sums of edge- labels, rather than differences of vertex labels. It is still not known, if every tree has a graceful labeling. In this paper, we are discussing about the edge graceful labeling of bicycle graphs and its uniqueness.