On additive vertex labelings - Christian Barrientos (Indonesian Journal of Combinatorics, 2020)
In a quite general sense, additive vertex labelings are those functions that assign nonnegative integers to the vertices of a graph and the weight of each edge is obtained by adding the labels of its end-vertices. In this work we study one of these functions, called harmonious labeling. We calculate the number of non-isomorphic harmoniously labeled graphs with n edges and at most n vertices. We present harmonious labelings for some families of graphs that include certain unicyclic graphs obtained via the corona product. In addition, we prove that all n-cell snake polyiamonds are harmonious; this type of graph is obtained via edge amalgamation of n copies of the cycle C3 in such a way that each copy of this cycle shares at most two edges with other copies. Moreover, we use the edge-switching technique on the cycle C4t to generate unicyclic graphs with another type of additive vertex labeling, called strongly felicitous, which has a solid bond with the harmonious labeling.