Article contents
A Problem on Edge-magic Labelings of Cycles
Published online by Cambridge University Press: 20 November 2018
Abstract
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
$n$.
Keywords
- Type
- Research Article
- Information
- 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.
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220418072500609-0743:S0008439500008122:S0008439500008122_inline01.png?pub-status=live)
- 5
- Cited by