K-Zumkeller graphs through mycielski transformation

Abstract

Let G = (V, E) be a simple graph. A 1-1 function f : V → ℕ , where ℕ is the set of natural numbers, is said to induce a k-Zumkeller graph G if the induced edge function f * : E → ℕ defined by f* (xy) = f (x) f (y) satisfies the following conditions: (i) f* (xy) is a Zumkeller number for every xy ∈ E. (ii) The total number of distinct Zumkeller numbers on the edges of G is k. A Mycielski transformation of a graph is a larger graph having more vertices and edges. In this article, the Mycielski transformation of a graphs such as path, cycle and star graphs have been computed and their k-Zumkeller graphs have been investigated by reducing the number of distinct Zumkeller numbers. AMS Subject Classification: 05C78