THE ENTIRE FACE IRREGULARITY STRENGTH OF A BOOK WITH POLYGONAL PAGES

Abstract

A face irregular entire labeling is introduced by Baca et al. recently, as a modification of the well-known vertex irregular and edge irregular total labeling of graphs and the idea of the entire colouring of plane graph. A face irregular entire k-labeling 𝜆:𝑉∪𝐸∪𝐹→{1,2,⋯,𝑘} of a 2-connected plane graph 𝐺=(𝑉,𝐸,𝐹) is a labeling of vertices, edges, and faces of 𝐺 such that for any two different faces 𝑓 and 𝑔, their weights 𝑤𝜆(𝑓) and 𝑤𝜆(𝑓) are distinct. The minimum 𝑘 for which a plane graph 𝐺 has a face irregular entire 𝑘-labeling is called the entire face irregularity strength of 𝐺, denoted by 𝑒𝑓𝑠(𝐺). This paper deals with the entire face irregularity strength of a book with 𝑚 𝑛-polygonal pages, where embedded in a plane as a closed book with 𝑛−sided external face.