A paradigmatic approach to investigate restricted hyper totient graphs

Raúl M. Falcón,Shahbaz Ali,Muhammad Khalid Mahmmod

doi:10.3934/math.2021223

Abstract

<abstract> Nowadays, the problem of finding families of graphs for which one may ensure the existence of a vertex-labeling and/or an edge-labeling based on a certain class of integers, constitutes a challenge for researchers in both number and graph theory. In this paper, we focus on those vertex-labelings whose induced multiplicative edge-labeling assigns hyper totient numbers to the edges of the graph. In this way, we introduce and characterize the notions of hyper totient graph and restricted hyper totient graph. In particular, we prove that every finite graph is a hyper totient graph and we determine under which assumptions the following families of graphs constitute restricted hyper totient graphs: complete graphs, star graphs, complete bipartite graphs, wheel graphs, cycles, paths, fan graphs and friendship graphs. </abstract>

Highlights

In 2017, Khalid and Shahbaz [1] introduced the notions of totient, super totient and hyper totient numbers
We say that a finite graph G = (V, E) is a restricted hyper totient graph (RHTG) if it is a simple HTG whose set of vertices may be labeled by the subset of positive integers {1, . . . , |V|}
We have introduced the notion of hyper totient graph as any finite graph G = (V, E) admitting an injective vertex-labeling f : V → N so that its induced multiplicative edge-labeling assigns a hyper totient number to each edge

Summary

Introduction

In 2017, Khalid and Shahbaz [1] introduced the notions of totient, super totient and hyper totient numbers (see [2]) Recall in this regard that a positive integer t is said to be totient if the sum of its co-prime residues is 2kt, with k ≥ 1. A positive integer h is said to be hyper totient if its set of co-prime residues including h can be divided into two nonempty disjoint subsets with equal sum. Of particular interest for the topic of this paper, it is remarkable the recent studies of Shahbaz and Khalid [1, 10] on graph labelings based on super totient numbers, and the introduction of both concepts of restricted super totient labeling and super totient index of graphs by Joshua and Wong [11].

Preliminaries

AIMS Mathematics

Hyper totient graphs

Characterization of families of restricted hyper totient graphs

Conclusion