Absolute mean graceful labeling of various \(m\)-shadow and \(m\)-splitting related graphs

Jaykeen Jani1, J. H. Jani1, V. J. Kaneria1
1Department of Mathematics, Saurashtra University, Rajkot – 360005, Gujarat, India

Abstract

A graph \(G=(V,E)\) is said to be an absolute mean graceful graph if there exists a one-to-one function \(f:V(G)\to \{0,\pm1,\pm2,\ldots,\pm|E(G)|\}\) such that the induced edge-labeling function \(f^*:E(G)\to \{1,2,\ldots,|E(G)|\}\), defined by \[f^*(xy)=\left\lceil{\dfrac{|f(x)-f(y)|}{2}}\right\rceil,\] is bijective. The labeling function \(f\) is called an absolute mean graceful labeling of the graph \(G\). In this paper, we obtain absolute mean graceful labelings for \(m\)-splitting and \(m\)-shadow graphs of various graphs.

Keywords: absolute mean graceful labeling, m-shadow graph, m-splitting graph