### Relaxing the injectivity constraint on arithmetic and harmonious labelings

#### Abstract

Several of the most studied graph labelings are injective functions, this constraint precludes some graphs from admitting such labelings; a well-known example is given by the family of trees that cannot be harmoniously labeled. In order to study the existence of these labelings for certain graphs, the injectivity constraint is often dropped. In this work we eliminate this condition for two different, but related, additive vertex labelings such as the harmonious and arithmetic labelings. The new labelings are called semi harmonious and semi arithmetic. We consider some families of graphs that do not admit the injective versions of these labelings, among the graphs considered here we have cycles and other cycle-related graphs, including the analysis of some operations like the Cartesian product and the vertex or edge amalgamation; in addition, we prove that all trees admit a semi harmonious labeling. Something similar is done with the concept of arithmetic labeling, studying finite unions of semi arithmetic graphs together with some general results.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2022.10.2.13

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