Properly even harmonious labelings of disconnected graphs
Joseph A. Gallian, Danielle Stewart · AKCE International Journal of Graphs and Combinatorics · 2015
A graph with edges is said to be harmonious if there is an injection from the vertices of to the group of integers modulo such that when each edge is assigned the label , the resulting edge labels are distinct. If is a tree, exactly one label may be used on two vertices. Over the years, many variations of harmonious labelings have been introduced.We study a variant of harmonious labeling. A function is said to be a properly even harmonious labeling of a graph with edges if is an injection from the vertices of to the integers from 0 to and the induced function from the edges of to defined by is bijective. This paper focuses on the existence of properly even harmonious labelings of the disjoint union of cycles and stars, unions of cycles with paths, unions of squares of paths, and unions of paths.