Γ-supermagic labeling of products of two cycles with dihedral groups

Dalibor Froncek

Abstract


A  Γ-supermagic labeling of a graph G=(V,E) is a bijection from E to a group Γ of order |E| such that for every vertex x∈V a product of labels of all edges incident with x is equal to the same element µ∈Γ.  A Γ-supermagic labeling of the Cartesian product of two cycles, CmℹCn for every m,n≥3 of the same parity was found recently [5, 6] for all Abelian groups of order 2mn. In this paper we present a Dk-supermagic labeling of the Cartesian, direct, and strong product by dihedral group Dk for any m,n≥3.

Keywords


Magic-type labeling, supermagic labeling, vertex-magic edge labeling, group supermagic labeling, product of cycles

Full Text:

PDF

DOI: http://dx.doi.org/10.19184/ijc.2025.9.1.1

References

D. Froncek, Supermagic labelings of Cn⁊Cm, (2022), https://arxiv.org/2212.14836

D. Froncek, Γ-supermagic labeling of 4-regular Archimedean graphs with dihedral groups, Bull. Inst. Combin. Appl., 104 (2025), 80--94.

D. Froncek, J. McKeown, J. McKeown, M. McKeown, ℤ2nm-supermagic labeling of Cn⁊Cm, Indones. J. Combin., 2(2) (2018), 57--71.

D. Froncek, M. McKeown, Note on diagonal construction of ℤ2nm-supermagic labeling of Cn⁊Cm, AKCE Int. J. Graphs Comb., 17(3) (2020), 952--954.

D. Froncek, P. Paananen, L. Sorensen, Group-supermagic labeling of Cartesian product of two odd cycles, Bull. Inst. Combin. Appl., 101 (2024), 39--53.

D. Froncek, P. Paananen, L. Sorensen, Group-supermagic labeling of Cartesian product of two even cycles, Discrete Math., 347(8) (2024), paper No. 113741.

J. Ivančo, On supermagic regular graphs, Math. Bohem., 125 (2000), 99--114.

P. Paananen, Γ-supermagic Labeling of Cn⁊Cm, MS Thesis, University of Minnesota Duluth, Duluth, MN, U.S.A., 2021.

J. Sedláček, Problem 27, in: M. Fiedler (Ed.), Theory of Graphs and Its Applications, Praha, 1964, pp.163–164.

L. Sorensen, Γ-supermagic Labeling of Cn⁊Cm, MS Thesis, University of Minnesota Duluth, Duluth, MN, U.S.A., 2020.

R. Stanley, Linear homogeneous diophantine equations and magic labelings of graphs, Duke Math. J., 40 (1973), 607--632.

R. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen-Macaulay rings, Duke Math. J., 43 (1976), 511--531.

B. M. Stewart, Magic graphs, Canad. J. Math., 18 (1996), 1031--1059.

P. M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc., 13 (1962), 47--52.


Refbacks

  • There are currently no refbacks.


ISSN: 2541-2205

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

View IJC Stats