Total Face Irregularity Strength of Certain Graphs

Abstract

The edge k-labeling ψ of G is defined by a mapping from EG to a set of integers 1,2,…,k, where the integer weight assigned to the vertex x∈VG is given as wψx=∑ψxy, such that the sum is taken over every vertex of y∈VG that is adjacent to x and the integer weights of adjacent vertices must be distinct for all vertices with x≠y. An irregular assignment of G using atmost k labels which is considered to be a minimum k is defined as irregularity strength of a graph G and can be denoted as sG. There are also further works on familiar irregular assignments, such as edge irregular labelings, vertex irregular total labelings, edge irregular total labelings, and face irregular entire k-labelings of plane graphs. A plane graph can be defined as a graph that is embedded in the plane in which no two lines will be intersected. In a plane graph the number of regions present are called faces and we denote it as F. The concept of total face irregularity strength is defined by the motivation of irregular networks and entire irregular face k-labeling. In our paper, we have obtained a minimum bound for the total face irregularity strength of two-connected plane graphs like cycle-of-ladder, C-necklace graph, P-necklace graph, sibling tree, and triangular graph.