Expand all Collapse all | Results 1 - 1 of 1 |
1. CMB 2013 (vol 57 pp. 375)
A Problem on Edge-magic Labelings of Cycles 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 Category:05C78 |