Irregularity and Modular Irregularity Strength of Wheels

Martin Bača,Andrea Semaničová-Feňovčíková,Muhammad Imran

doi:10.3390/math9212710

Martin Bača, Andrea Semaničová-Feňovčíková + Show 1 more

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https://doi.org/10.3390/math9212710

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Abstract

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel edges required to ensure that all vertices have distinct degrees? This problem is known as the graph labeling problem and, for its solution, Chartrand et al. introduced irregular assignments. The irregularity strength of a graph G is known as the maximal edge label used in an irregular assignment, minimized over all irregular assignments. Thus, the irregularity strength of a simple graph G is equal to the smallest maximum multiplicity of an edge of G in order to create an irregular multigraph from G. In the present paper, we show the existence of a required irregular labeling scheme that proves the exact value of the irregularity strength of wheels. Then, we modify this irregular mapping in six cases and obtain labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength.

Highlights

Given a function ψ : E(G) → {1, 2, . . . , k}, the weight of a vertex x is wtψ(x) = ∑y∈N(x) ψ(xy), where N(x) denotes the set of neighbors of x in G. Such a function ψ we call an irregular assignment if wtψ(x) = wtψ(y) for all vertices x, y ∈ V(G) with x = y
The irregularity strength s(G) of a graph G is known as the maximal integer k, minimized over all irregular assignments, and is set to ∞ if no such function is possible
We investigated the existence of an irregular assignment of wheels and we determined the exact value of the irregularity strength of wheels Wn of order n + 1, n ≥ 3, as follows: s(Wn) =

Summary

Introduction

K}, the weight of a vertex x is wtψ(x) = ∑y∈N(x) ψ(xy), where N(x) denotes the set of neighbors of x in G Such a function ψ we call an irregular assignment if wtψ(x) = wtψ(y) for all vertices x, y ∈ V(G) with x = y. We improve the main idea of the construction of an irregular assignment for fan graphs used in [13] and we construct an edge labeling with the desired irregular properties. The existence of such labeling proves the exact value of the irregularity strength of wheels. We modify this irregular mapping of wheels in six cases, and, for each case, we determine the exact value of the modular irregularity strength

Irregular Assignment of Wheels

Modular Irregular Assignment of Wheels

Conclusions