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ODD GRACEFUL LABELING OF ARBITRARY SUPERSUBDIVISION OF CERTAIN GRAPHS
Keywords:
arbitrary supersubdivision, odd graceful labeling, path, caterpillar, shell graph.DOI:
https://doi.org/10.17654/0974165822041%20Abstract
An odd graceful labeling of a graph $G$ with $q$ edges is an injection $f$ from $V(G)$ to $\{0,1,2, \ldots, 2 q-1\}$ such that when an edge $x y$ is assigned the label $|f(x)-f(y)|$, then the resulting edge labels are $\{1,3,5, \ldots, 2 q-1\}$. A graph $H$ is called a supersubdivision graph of a graph $G$ if $H$ is obtained from $G$ by replacing every edge $u v$ of $G$ by a complete bipartite graph $K_{2, m}$ ( $m$ may vary for each edge) and identifying $u$ and $v$ with the two vertices in $K_{2, m}$ that form one of the two partite sets. In this paper, we prove that the graphs obtained by the arbitrary supersubdivision of path, caterpillar and shell graphs are odd graceful.
Received: May 30, 2022
Accepted: July 15, 2022
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