M-gracefulness of graphs

Abstract

Let G = (V,E) be a (p, q)-graph without isolated vertices. The gracefulness grac(G) of G is the smallest positive integer k for which there exists an injective function f: V → {0, 1, 2,..., k} such that the edge induced function gf: E → {1, 2,..., k} defined by gf (uv) = |f(u)− f(v)|, ∀uv ∈ E is also injective. Let c(f) = max{i: 1, 2,..., i are edge labels} and let m(G) = maxf {c(f)} where the maximum is taken over all injective functions f: V → ℕ ∪ {0} such that gf is also injective. This new measure m(G) is called m-gracefulness of G and it determines how close G is to being graceful. In this paper, we prove that there are infinitely many nongraceful graphs with m-gracefulness q − 1, we give necessary conditions for a (p, q)-eulerian graph and the complete graph Kp to have m-gracefulness q −1 and q −2. Using this, we prove that K5 is the only complete graph to have m-gracefulness q−1. We also give an upper bound for the highest possible vertex label of Kp if m(Kp) = q−2.