Super Vertex-magic Total Labelings of \(W_{3,n}\)

Abstract

A graph \(G\) is called super vertex-magic total labelings if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1,2,\ldots,|V(G)| + |E(G)|\}\) such that \(f(v) + \sum_{u \sim v} f(vu) = C\), where the sum is over all vertices \(u\) adjacent to \(v\) and \(f(V(G)) = \{1,2,\ldots,|V(G)|\}\), \(f(E(G)) = \{|V(G)|+1,|V(G)|+2,\ldots,|V(G)|+|E(G)|\}\). \({The Knödel graphs}\) \(W_{\Delta,n}\) have even \(n \geq 2\) vertices and degree \(\Delta\), \(1 \leq \Delta \leq \lfloor\log_2 n\rfloor\). The vertices of \(W_{\Delta,n}\) are the pairs \((i,j)\) with \(i = 1,2\) and \(0 \leq i \leq n/2-1\). For every \(j\), \(0 \leq j \leq n/2-1\), there is an edge between vertex \((1,j)\) and every vertex \((2,(j+2^k-1) \mod (n/2))\), for \(k=0,\ldots,\Delta-1\). In this paper, we show that \(W_{3,n}\) is super vertex-magic for \(n \equiv 0 \mod 4\).