A Problem on Edge-magic Labelings of Cycles

Abstract

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 :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 ≥ 7. Motivated by this conjecture, we prove that for all n 0 ∈ N, there exists n ∈ 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.

New Constructions for the n-Queens Problem

“…Note that when h is constant, D ⊗ h Γ is the Kronecker product. Many relations among labelings have been established using the ⊗ h -product and some particular families of graphs, namely S p and S k p (see for instance, [11,13,14,15,16,17,18]). The family S p contains all super edge-magic 1-regular labeled digraphs of order p where each vertex takes the name of the label that has been assigned to it.…”

Perfect (super) Edge-Magic Crowns

“…The corona product of two graphs G and H is the graph G ⊙ H obtained by placing a copy of G and |V (G)| copies of H and then joining each vertex of G with all vertices in one copy of H in such a way that all vertices in the same copy of H are joined exactly to one vertex of G. Let K n be the complementary graph of the complete graph K n , n ∈ N. Theorem 1.1. [12,13] Let C m be a cycle of order m = p k , where p > 2 is a prime number. Then the graph G ∼ = C m ⊙ K n is perfect (super) edge-magic.…”

“…[13] Let G be a (p, q)-graph with p = q and let f :V (G) ∪ E(G) → {i} p+q i=1be a super edge-magic labeling of G. Then, the odd labeling o(f ) and the even labeling e(f ) obtained from f are edge-magic labelings of G with valences val(o(f )) = 2val(f ) − 2p − 2 and val(e(f )) = 2val(f ) − 2p − 1 respectively.…”

(Di)graph Decompositions and Magic Type Labelings: A Dual Relation

“…Many relations among labelings have been established using the ⊗ h -product and some particular families of graphs, namely S p and S k p (see for instance, [11,16,18,20]). The family S p contains all super edge-magic 1-regular labeled digraphs of order p where each vertex takes the name of the label that has been assigned to it.…”