Abstract
A rooted graph is a pair (G,x), where G is a simple undirected graph and x β V(G). If G is rooted at x, its k__th rotation number h~k~__ (G,x) is the minimum number of edges in a graph F of order |G| + k such that for every v β V(F) we can find a copy of G in F with the root vertex x at v. When k = 0, this definition reduces to that of the rotation number h(G,x), which was introduced in [βOn Rotation Numbers for Complete Bipartite Graphs,β University of Victoria, Department of Mathematics Report No. DMβ186βIR (1979)] by E.J. Cockayne and P.J. Lorimer and subsequently calculated for complete multipartite graphs. In this paper, we estimate the __k__th rotation number for complete bipartite graphs G with root x in the larger vertex class, thereby generalizing results of B. BollobΓ‘s and E.J. Cockayne [βMore Rotation Numbers for Complete Bipartite Graphs,β Journal of Graph Theory, Vol. 6 (1982), pp. 403β411], J. Haviland [βCliques and Independent Sets,β Ph. D. thesis, University of Cambridge (1989)], and J. Haviland and A. Thomason [βRotation Numbers for Complete Bipartite Graphs,β Journal of Graph Theory, Vol. 16 (1992), pp. 61β71]. Β© 1993 John Wiley & Sons, Inc.