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A Problem on Edge-magic Labelings of Cycles

Published online by Cambridge University Press:  20 November 2018

S. C. López
Affiliation:
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, BarcelonaTech, C/Esteve Terrades 5, 08860 Castelldefels, Spain e-mail: [email protected]@gmail.com
F. A. Muntaner-Batle
Affiliation:
Graph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of Newcastle, NSW 2308 Australia e-mail: [email protected]
M. Rius-Font
Affiliation:
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, BarcelonaTech, C/Esteve Terrades 5, 08860 Castelldefels, Spain e-mail: [email protected]@gmail.com
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Abstract

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In 1970, Kotzig and Rosa defined the concept of edge-magic labelings as follows. Let $G$ be a simple $\left( p,\,q \right)$-graph (that is, a graph of order $p$ and size $q$ without loops or multiple edges). A bijective function $f:\,V\left( G \right)\cup E\left( G \right)\,\to \,\left\{ 1,\,2,\,.\,.\,.\,,\,p\,+\,q \right\}$ is an edge-magic labeling of $G$ if $f\left( u \right)\,+\,f\left( uv \right)\,+f\left( v \right)\,=\,k$, for all $uv\,\in \,E\left( G \right)$. A graph that admits an edge-magic labeling is called an edge-magic graph, and $k$ is called the magic sum of the labeling. An old conjecture of Godbold and Slater states that all possible theoretical magic sums are attained for each cycle of order $n\,\ge \,7$. Motivated by this conjecture, we prove that for all ${{n}_{0}}\,\in \,\mathbb{N}$, there exists $n\,\in \,\mathbb{N}$ such that the cycle ${{C}_{n}}$ admits at least ${{n}_{0}}$ edge-magic labelings with at least ${{n}_{0}}$ mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle ${{C}_{n}}$, depending on the sum of the exponents of the odd primes appearing in the prime factorization of $n$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

The first and the third author are supported by the Spanish Research Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under grant 2009SGR1387.

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