On the graceful numbering of some classes of trees

Abstract

The graceful numbering, one of the vertex numberings, is considered. The numbering is described as follows; Given a connected graph G(V,E) with $\mid{E}\mid$=e, we wish to assign distinct nonnegative integers to the vertices so that the edge numbers form the set {1,2,$\dots$,e}, where each edge number is defined to be the absolute difference between the numbers assigned to the end vertices of the edge. Moreover, we wish to minimize the maximum value of the vertex numbers. We call this minimized value g(G). If g(G)=e, then G is called to be a ``graceful graph,`` and the numbering achieving g(G)=e, a ``graceful numbering.`` Several workers have extended the area of graceful trees since Ringel conjectured that all trees are graceful. In this thesis, we shall introduce a sufficient condition for trees to be graceful, and, using it, show that two large classes of trees, ``M-trees`` including full m-ary trees, m≥1, and ``Z-routable trees,`` are graceful.