ON THE SUPER EDGE-MAGIC DEFICIENCY AND α-VALUATIONS OF GRAPHS
Abstract: A graph G is called
super edge-magic if there exists a bijective function f : V (G) [ E (G) ! f1;
2; : : : ; jV (G)j + jE (G)jg such that f (u) + f (v) + f (uv)is a constant for
each uv 2 E (G) and f (V (G)) = f1; 2; : : : ; jV (G)jg. The superedge-magic
deficiency, µs (G), of a graph G is defined as the smallest nonnegativeinteger
n with the property that the graph G [ nK1 is super edge-magic or +1 ifthere
exists no such integer n. In this paper, we prove that if G is a graph withoutisolated
vertices that has an α-valuation, then µs (G) ≤ jE (G)j − jV (G)j + 1. This leads
to µs (G) = jE (G)j − jV (G)j + 1 if G has the additional property that G is not
sequential. Also, we provide necessary and sufficient conditions for the
disjoint union of isomorphic complete bipartite graphs to have an α-valuation.
Moreover,we present several results on the super edge-magic deficiency of the
same class ofgraphs. Based on these, we propose some open problems and a new
conjecture.
Key words: Super edge-magic
labeling, super edge-magic deficiency, sequential labeling, sequential number,
α-valuation
Author: Rikio Ichishima and
Akito Oshima
Journal Code: jpmatematikagg110029
