The total face irregularity strength of some plane graphs

Abstract

A face irregular total k-labeling λ : V ∪ E → {1, 2, . . . , k} of a 2-connected plane graph G is a labeling of vertices and edges such that their face-weights are pairwise distinct. The weight of a face f under a labeling λ is the sum of the labels of all vertices and edges surrounding f . The minimum value k for which G has a face irregular total k-labeling is called the total face irregularity strength of G, denoted by t f s(G). The lower bound of t f s(G) is provided along with the exact value of two certain plane graphs. Improving the results, this paper deals with the total face irregularity strength of the disjoint union of multiple copies of a plane graph G. We estimate the bounds of t f s(G) and prove that the lower bound is sharp for G isomorphic to a cycle, a book with m polygonal pages, or a wheel.

Total Face Irregularity Strength of Certain Graphs

“…In 2022, Bača et al [12] investigated the irregular labelings with respect to face of plane graphs and have found a new graph characteristic which can be termed as face irregularity strength of a few types ðα; β; γÞ: . Also, Tilukay et al [13] have estimated the bounds of total face irregularity strength tfsðGÞ: and have proved that the lower bound is sharp for G isomorphic to a cycle, a book with m polygonal pages, or a wheel. Further, Jamil and Mughal [14] have studied tfs of generalized plane grid graphs G n m and wheel graphs W n under a graph k-labeling of type ðα; β; γÞ: where α; β 2 0; 1.…”