A comprehensive survey on 3-equitable and divisor 3-equitable labeling of graphs

This article presents a short survey on 3-equitable and divisor 3-equitable labeling of graphs. For any graph G ( V , E ) and k > 0, assign vertex labels from {0,1,…, k − 1} such that when the edge labels induced by the absolute value of the difference of the vertex labels, the number of vertices labeled with i and the number of vertices labeled with j differ by at most one and the number of edges labeled with i and the number of edges labeled with j differ by at most one. We call a graph G with such an assignment of labels k –equitable. When k = 3, it becomes a 3-equitable labeling. In 2019, Sweta Srivastav et al. introduced the notion of divisor 3-equitable labeling of graphs. A bijection f : V ( G ) → {1, 2, …, n } induces a function f ': E ( G ) → {0,1,2} defined by for each edge e = xy , (i) f '( e ) = 1 if f ( x )| f ( y ) or f ( y )| f ( x ), (ii) f '( e ) = 2 if f ( x )/ f ( y ) = 2 or f ( y )/ f ( x ) = 2, and (iii) f '( e ) = 0 otherwise such that | e f ′ ( i ) − e f ′ ( j ) | ≤ 1 for all 0 ≤ i,j ≤ 2. A graph which admits a divisor 3-equitable labeling is called a divisor 3-equitable graph. This article stands divided into five sections. The first and fifth sections are reserved respectively for introduction and some important references. The second section deals with the 3-equitable labeling of graphs wherein some important known results have been recalled. The third section deals with the divisor 3-equitable labeling of graphs wherein a few known results have been outlined. In the fourth section we highlight certain conjectures and open problems in respect of the above mentioned labeling that still remain unsolved.