Irregularity and Modular Irregularity Strength of Wheels

Martin Bača, Muhammad Imran and Andrea Semaničová-Feňovčíková
Additional contact information
Martin Bača: Department of Applied Mathematics and Informatics, Technical University, 042 00 Košice, Slovakia
Muhammad Imran: Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, United Arab Emirates
Andrea Semaničová-Feňovčíková: Department of Applied Mathematics and Informatics, Technical University, 042 00 Košice, Slovakia

Mathematics, 2021, vol. 9, issue 21, 1-14

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.

Keywords: irregular assignment; (modular) irregularity strength; wheel (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/9/21/2710/pdf (application/pdf)
https://www.mdpi.com/2227-7390/9/21/2710/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:9:y:2021:i:21:p:2710-:d:664311

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().