Super edge-gracefulness of complete graphs
Abstract
A graph G(V, E) of order | V| = p and size |E| = q is called super edge-graceful if there is a bijection f from E to {0, ±1,..., ±q-1/2} when q is odd and from E to {±1,..., ±q/2} when q is even such that the induced vertex labeling f * defined by f *(x) = Σ xυ∈e(G) f (xy) over all edges xy is a bijection from V to {0, ±1, ±2..., ±p-1/2} when p is odd and from V to {±1, ±2,..., ±p/2} when p is even. Sin-Min Lee, Ling Wang and Emmanuel R. Yera (Congressus Numerantium 174(2005) 83-96) posed the following problem: For which n, the complete graph K n is super edge-graceful? It is known that the complete graphs K n for n = 3,5,6,7,8 are super edge-graceful and K 4 is not super edge-graceful. In this paper we prove that all complete graphs of order n ≥ 3, n ≠ 4, are super edge-graceful.
