Characterizing Irregularity in Planar Graph Structures

Abdul Aleem Mughal, Raja Noshad Jamil, Abaid ur Rehman Virk \(^{∗,∗}\)

\(^a\)University of Management and Technology, Lahore, Pakistan.

\(^b\)University of Management and Technology, Lahore, Pakistan

Correspondence should be addressed to: f2016265007@umt.edu.pk, noshad.jamil@yahoo.com, abaidrehman@umt.edu.pk

Copyright © 2024 Abdul Aleem Mughal, Raja Noshad Jamil, Abaid ur Rehman Virk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: 19 October 2023 Accepted: 11 January 2024; Published Online: 15 January 2024.

Abstract

Face irregularity strength under ρ−labeling ξ with class (α1, β1, γ1) of plane graphs is a labeling from the set of graph elements into the set of integers, that is, ξ: {V ∪ E ∪ F} → {1, 2, 3, .., ρ}, such that the face weights are distinct at any stage in the graph labeling, that is, Wξ(α1,β1,γ1)(f) ̸= Wξ(α1,β1,γ1)(g), for any two faces f and g of the graph G. The face irregular strength of a plane graph G is the least possible integer ρ such that G admits face irregular ρ−labeling. In this research, authors have examined the exact tight lower bounds for the face irregular strength of generalized plane graphs under ρ−labeling of class (α1, β1, γ1) for vertex (1, 0, 0), edge (0, 1, 0), face (0, 0, 1), vertex-face (1, 0, 1), edge-face (0, 1, 1) and entire (1, 1, 1). Results are verified by examples.

Keywords:

Cartesian product, Face labeling, Face irregularity strength, Face weights.