Super Edge-Graceful Labelings of Complete Tripartite Graphs

Let \( [n]^* \) denote the set of integers \(\{-\frac{n-1}{2}, \ldots, \frac{n+1}{2}\}\) if \( n \) is odd, and \(\{-\frac{n}{2}, \ldots, \frac{n}{2}\} \setminus \{0\}\) if \( n \) is even. A super edge-graceful labeling \( f \) of a graph \( G \) of order \( p \) and size \( q \) is a bijection \( f : E(G) \to [q]^* \), such that the induced vertex labeling \( f^* \) given by \( f^*(u) = \sum_{uv \in E(G)} f(uv) \) is a bijection \( f^* : V(G) \to [p]^* \). A graph is super edge-graceful if it has a super edge-graceful labeling. We prove that all complete tripartite graphs \( K_{a,b,c} \), except \( K_{1,1,2} \), are super edge-graceful.