Vertex-Magic Total Labeling of the Union of Suns

Abstract

Let \(G\) be a graph with vertex-set \(V = V(G)\) and edge-set \(E = E(G)\), and let \(e = |E(G)|\) and \(v = |V(G)|\). A one-to-one map \(\lambda\) from \(V \cup E\) onto the integers \(\{1, 2, \ldots, v+e\}\) is called a vertex-magic total labeling if there is a constant \(k\) so that for every vertex \(x\),

\[\lambda(x) + \sum \lambda(xy) = k\]

where the sum is over all edges \(xy\) where \(y\) is adjacent to \(x\). Let us call the sum of labels at vertex \(x\) the weight \(w_\lambda\) of the vertex under labeling \(\lambda\); we require \(w_\lambda(x) = k\) for all \(x\). The constant \(k\) is called the magic constant for \(\lambda\).

A sun \(S_n\) is a cycle on \(n\) vertices \(C_n\), for \(n \geq 3\), with an edge terminating in a vertex of degree \(1\) attached to each vertex.

In this paper, we present the vertex-magic total labeling of the union of suns, including the union of $m$ non-isomorphic suns for any positive integer $m \geq 3$, proving the conjecture given in [6].