On $k$-layered numbers and some labeling related to $k$-layered numbers

Abstract

In this paper, first, we define and investigate $k$-layered numbers, which are a generalization of Zumkeller numbers. After that, we generalize the concept of Zumkeller labeling and Zumkeller cordial labeling to $k$-layered labeling and $k$-layered cordial labeling, respectively. Moreover, we prove that every simple graph admits Zumkeller labeling, Zumkeller cordial labeling, 3-layered labeling, 3-layered cordial labeling, 4-layered labeling and 4-layered cordial labeling.

There exists some typo mistakes in some of proofs