Two Rosa-type Labelings of Uniform k-distant Trees and a New Class of Trees

A k-distant tree consists of a main path, called the spine, such that each vertex on the spine is joined by an edge to an end-vertex of at most one path on at most k vertices. Those paths, along with the edge joining them to the spine, are called tails. When every vertex on the spine has exactly one incident tail of length k we call the tree a uniform k-distant tree. We show that every uniform k-distant tree admits both a graceful- and an α-labeling. For a graph G and a positive integer a, define appa(G) to be the graph obtained from appending a leaves to each leaf in G. When G is a uniform k-distant tree, we show that appa(G) admits both a graceful- and an α-labeling