Harmonious Labelings Via Cosets and Subcosets

In [4], it is shown that the disjoint union of an odd cycle and certain paths is harmonious, and that certain starlike trees are harmonious using properties of cosets for a particular subgroup of Z m , where m is the number of edges of the graph. We expand upon these results by first exploring the numerical properties when adding values from cosets and subcosets in Z m . We will then show that these properties may be used to harmoniously label graphs involving a more complex starlike tree, which we will call the snowflake graph.