I have a problem involving graph theory. This is not a graph theory course though so I do not know much about it. If we have complete bipartite graphs consider whethere or not they are graceful. A graph is graceful provided the vertices of the graph can be assigned v distinct integers from 0 to e in such a way that each integer from 1 to e arises as the absolute value of the difference of the numbers assigned to adjacent vertices. I think I have read somewhere that a bipartite graph has an odd graceful labeling. I am not sure what this means or if this is even correct. But i need a make some kind of conjecture about complete bipartite graphs and whether or not they are graceful. Also another question would be given a graceful, bipartite graph it is sometimes possible to modify the grapg so as to produce a graceful, nonbipartite graph with the same number of edges and one fewer vertices, when and how? Does anoyone have ideas about this questions?