A merging procedure for labelings of bipartite graphs

Let $G$ a bipartite graph with vertex bipartition $\{A,B\}$ and let $m=\mathrm{\mid }E(G)\mathrm{\mid }$. An $(A,B)$-uniformly ordered labeling of $G$ is a labeling $f\text{\hspace{0.17em}⁣}:V\to [0,2m]$ which, among other conditions, requires that there exists $\lambda \in \mathbb{N}$ such that $f(a)\le \lambda$ and $f(b)>\lambda$ for all $a\in A$ and $b\in B$. The existence of such a labeling for $G$ implies the existence of a cyclic $G$-decomposition of $K_{2mx+1}$ for all positive integers $x$. In this paper, as a starting point, through this type of labeling we prove the existence of a cyclic $G$-decomposition in the case that $G$ is a cycle of even length with either one or two pendant paths of any length. Then, through a merging procedure, we are able to get this type of labeling for a specific class of bipartite graphs, which are obtained by iteratively adding an even cycle and a pendant path.