Irregularity and Modular Irregularity Strength of Wheels

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 mult...

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Autores principales:	Martin Bača, Muhammad Imran, Andrea Semaničová-Feňovčíková
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	irregular assignment (modular) irregularity strength wheel Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/0f495038f2d44c91a2975a68b469df26
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spelling	oai:doaj.org-article:0f495038f2d44c91a2975a68b469df262021-11-11T18:16:11ZIrregularity and Modular Irregularity Strength of Wheels10.3390/math92127102227-7390https://doaj.org/article/0f495038f2d44c91a2975a68b469df262021-10-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2710https://doaj.org/toc/2227-7390It 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 <i>G</i> 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 <i>G</i> is equal to the smallest maximum multiplicity of an edge of <i>G</i> in order to create an irregular multigraph from <i>G</i>. 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.Martin BačaMuhammad ImranAndrea Semaničová-FeňovčíkováMDPI AGarticleirregular assignment(modular) irregularity strengthwheelMathematicsQA1-939ENMathematics, Vol 9, Iss 2710, p 2710 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	irregular assignment (modular) irregularity strength wheel Mathematics QA1-939
spellingShingle	irregular assignment (modular) irregularity strength wheel Mathematics QA1-939 Martin Bača Muhammad Imran Andrea Semaničová-Feňovčíková Irregularity and Modular Irregularity Strength of Wheels
description	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 <i>G</i> 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 <i>G</i> is equal to the smallest maximum multiplicity of an edge of <i>G</i> in order to create an irregular multigraph from <i>G</i>. 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.
format	article
author	Martin Bača Muhammad Imran Andrea Semaničová-Feňovčíková
author_facet	Martin Bača Muhammad Imran Andrea Semaničová-Feňovčíková
author_sort	Martin Bača
title	Irregularity and Modular Irregularity Strength of Wheels
title_short	Irregularity and Modular Irregularity Strength of Wheels
title_full	Irregularity and Modular Irregularity Strength of Wheels
title_fullStr	Irregularity and Modular Irregularity Strength of Wheels
title_full_unstemmed	Irregularity and Modular Irregularity Strength of Wheels
title_sort	irregularity and modular irregularity strength of wheels
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/0f495038f2d44c91a2975a68b469df26
work_keys_str_mv	AT martinbaca irregularityandmodularirregularitystrengthofwheels AT muhammadimran irregularityandmodularirregularitystrengthofwheels AT andreasemanicovafenovcikova irregularityandmodularirregularitystrengthofwheels
_version_	1718431865982943232

Irregularity and Modular Irregularity Strength of Wheels

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