Total Face Irregularity Strength of Certain Graphs
D. Ahima Emilet, Daniel Paul, R. Jayagopal, Micheal Arockiaraj · Mathematical Problems in Engineering · 2024
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.