Total Face Irregularity Strength of Certain Graphs

Abstract

The edge $k$-labeling $\psi$ of $G$ is defined by a mapping from $E\left(G\right)$ to a set of integers $\left\{1,2,\dots ,k\right\}$, where the integer weight assigned to the vertex $x\in V\left(G\right)$ is given as $w_{\psi }\left(x\right)=\sum \psi \left(xy\right)$, such that the sum is taken over every vertex of $y\in V\left(G\right)$ that is adjacent to $x$ and the integer weights of adjacent vertices must be distinct for all vertices with $x\ne y$. 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 $s\left(G\right)$. 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.