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 tfs(G). The lower bound of tfs(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 tfs(G) and prove that the lower bound is sharp for G isomorphic to a cycle, a book with m polygonal pages, or a wheel.