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# A Problem on Edge-magic Labelings of Cycles

Published:2013-10-12
Printed: Jun 2014
• S. C. López,
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya. BarcelonaTech, C/Esteve Terrades 5, 08860 Castelldefels, Spain
• Muntaner-Batle,
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
• Rius-Font,
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya. BarcelonaTech, C/Esteve Terrades 5, 08860 Castelldefels, Spain
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## Abstract

Kotzig and Rosa defined in 1970 the concept of edge-magic labelings as follows: let $G$ be a simple $(p,q)$-graph (that is, a graph of order $p$ and size $q$ without loops or multiple edges). A bijective function $f:V(G)\cup E(G)\rightarrow \{1,2,\ldots,p+q\}$ is an edge-magic labeling of $G$ if $f(u)+f(uv)+f(v)=k$, for all $uv\in E(G)$. 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 sets 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: edge-magic, valence, $\otimes_h$-product
 MSC Classifications: 05C78 - Graph labelling (graceful graphs, bandwidth, etc.)