On Super Vertex-Magic Total Labeling of the Disjoint Union of k Copies of K_n

Another paper on VMT labelings. General construction of several VMT labelings for disjoint union of k Copies of K_n when n>4.

Abstract

Let G=(V,E) be a finite non-empty graph. A vertex-magic total labeling (VMTL) is a bijection λ from V \cup E to the set of consecutive integers { 1, 2, ..., |V|+|E| } with the property that for every v∈V, λ(v) +∑_w∈N(v) λ(vw) = h, for some constant h. Such a labeling is called super if the vertex labels are 1, 2, ..., |V|.

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 = 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 = 2 (mod 4). Let n = 4l+2 for a positive integer l. The graph k K_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.

Status

Accepted for publication in Ars Combinatoria.
P.Kovář, J. Gómez: On Super Vertex-Magic Total Labeling of the Disjoint Union of k Copies of K_n, accepted (In Ars Combinatoria).

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Last update: 29.12.2011