Proper lucky labelling for mӧbius ladder graphs

Abstract

Proper Lucky labelling is a graph labelling concept in which adjacent vertices are assigned distinct labels, and the sums of the labels of adjacent vertices are also distinct. A graph that satisfies both of these conditions is called a Proper Lucky labelling graph. The smallest natural number needed to label the graph in this manner is known as the Proper Lucky number, denoted by η(p)(G). The Mӧbius ladder graph (M₂(n)) is a graph formed by connecting two cycles with edges in a twisted pattern, creating a non-orientable structure. This study focuses on determining the Proper Lucky number of the Möbius ladder graph and deriving a general formula for labelling this class of graphs. In Proper Lucky labelling, vertex labelling should be from the set {1,2,3, ... , k} such that k is the least natural number that labels the graph, and gets general formulas for vertex labelling of each graph. For that, we consider n number of vertices in one cycle, and we get 2 as the Proper Lucky number for the Möbius ladder graph when n is odd, otherwise, it is 4. Further, it is proved that these general formulas are valid for infinite graphs. Future studies can explore the Proper Lucky labelling of generalized Möbius like structures and other non-orientable or topologically complex graphs. Also, algorithmic approaches could be explored to compute Proper Lucky numbers for large-scale or dynamic graphs.