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 \(V \cup E\) to the set of consecutive integers \(\{1, 2, \ldots, |V| + |E|\}\) with the property that for every \(v \in V\), \(\lambda(v) + \sum_{w \in N(v)} \lambda(vw) = h\), for some constant \(h\). Such a labeling is called super if the vertex labels are \(1, 2, \ldots, |V|\).
There are some results known about super VMTLs 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 \(4 < n \equiv 0 \pmod{4}\) and any \(k\), and for \(n = 4\) and \(k\) even. We continue the study and examine the graph \(kK_n\) for \(n \equiv 2 \pmod{4}\). 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 VMTLs of \(4K_{2l+1}\).