Graceful labelings and graceful orientations of graphs

A graph with m edges is defined to be graceful if its vertices can be labeled using the integers 0,1,...,m which are distinct from each other, such that the absolute values of the differences between adjacent labels are precisely the integers 1,2,...,m.This study introduces a new class of cyclic graph called flowerette and characterizes those that are graceful. Further, a 3-regular bipartite graceful graph is obtained from flowerettes. It is also shown that the gracefulness/non-gracefulness of graphs obtained from binary operations is not dependent on the gracefulness/non-gracefulness of the graphs involved. Bloom and Hsu extended the concept of graceful graphs to digraphs. Likewise, Gervacio introduced the concept of residually graceful digraphs. In this research, graceful and residually graceful labelings of the oriented star Sm and those of the oriented flowerettes Ftn are investigated