Harmonious Properties of Uniform -Distant Trees

We prove that all uniform -distant trees are harmonious; every uniform -distant odd tree is strongly -harmonious, so is every uniform -distant even tree if the spine has even number of vertices. Also, all uniform -distant trees are sequential. 1. Introduction Labeling of a graph is an assignment of labels (numbers) to its vertices or/and edges or faces, which satisfy some conditions. These are different from coloring problems since some properties and structures of numbers such as ordering, addition, and subtraction used here are not properties of colors. Graph labelings have several applications in many fields. They have found usage in various coding theory problems, including the design of good radar-type codes, synch-set codes, and convolutional codes with optimal autocorrelation properties. They facilitate the optimal nonstandard encodings of integers. They have also been applied to determine ambiguities in X-ray crystallographic analysis, to design of a communication network addressing system, to determine optimal circuit layouts, and to problems in additive number theory. Graham and Sloane [1] have introduced harmonious graphs in their study of modular versions of additive bases problems stemming from error-correcting codes. The conjecture “All Trees are Harmonious” is still open and is unsettled for many years. Gallian in his survey [2] of graph labeling has mentioned that no attention has been given to analyze the harmonious property of lobsters. It is clear that uniform 2-distant trees are special lobsters. Murugan [3] has proved that uniform -distant trees are graceful and have many interesting properties. Also, Abueida and Roberts [4] have proved that uniform -distant trees admit a harmonious labeling, when they have even number of vertices. In this paper, we prove that all uniform -distant trees are harmonious; and every uniform -distant odd tree is strongly -harmonious; every uniform -distant even tree is strongly -harmonious, when the spine has even number of vertices. Also, all uniform -distant trees are sequential. 2. -Distant Tree A -distant tree consists of a main path called the “spine,” such that each vertex on the spine is joined by an edge to at most one path on -vertices. Those paths are called “tails” (i.e., each tail must be incident with a vertex on the spine). When every vertex on the spine has exactly one incident tail of length , we call the tree a uniform -distant tree. A uniform -distant tree with odd number of vertices is called a uniform -distant odd tree. A uniform -distant tree with even number of vertices is called a