Odd graceful labeling of \(W\)-tree \(WT(n, k)\) and its disjoint union (Q6559882)

The concept of odd graceful labeling was introduced in the year 1991 [\textit{R. B. Gnanajothi}, Topics in graph theory. Madurai: Madurai Kamaraj University (PhD Thesis) (1991)] and thereafter several results have been published by various authors on this topic. This is yet another study on odd graceful labeling and the authors present the study in four sections. In the first section, they give a brief literature review, definitions, and the terminology used by them in this manuscript. In the next section, they establish that the $W$-tree admits odd graceful labeling for $k\geq 3$ and $n\geq 2k-2$. Section three contains odd graceful labeling of the disjoint union of two isomorphic copies of $W$-tree $WT(n,k)$ for $k\geq 3$ and $n\geq 4k-2$. In the final section, they exhibit the odd graceful labeling of the disjoint union of the W-tree with other graphs, such as the path, star, bistar, and ladder. Note that Figure 1 illustrates the $W$-tree graph $WT(3,3)$ and Figure 2 illustrates the odd graceful labeling of $W$-tree graph $WT(4,3)$.