On super vertex-magic total labeling of the disjoint union of k copies of K-n

Abstract

Let G = (V, E) be a finite non-empty graph. A vertex-magic total labeling (VMTL) is a bijection lambda from V boolean OR E to the set of consecutive integers {1, 2,..., vertical bar V vertical bar + vertical bar E vertical bar} with the property that for every v is an element of V, lambda(v) + Sigma w is an element of N(v) lambda(vw) = h, for some constant h. Such a labeling is called super if the vertex labels are 1, 2,..., vertical bar V vertical bar

There are some results known about super VMTL of kG only when the graph G has a super VMTL. In this paper we focus on the case when G is the complete graph K-n. It was shown that a super VMTL of kK(n) exists for n odd and any k, for 4 < n equivalent to 0 (mod 4) and any k, and for n = 4 and k even. We continue the study and examine the graph kK(n) for n equivalent to 2 (mod 4). Let n = 4l + 2 for a positive integer l. The graph kK(4l+2) does not admit a super VMTL for k odd. We give a large number of super VMTLs of kK(4l+2) for any even k based on super VMTL of 4K(2l+1).