Super Connectivity and Super Arc Connectivity of the Mycielskian of a Digraph

Abstract

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph \(G\) into a new graph \(\mu(G)\), which is called the Mycielskian of \(G\).This paper shows that:
For a strongly connected digraph \(D\) with \(|V(D)| \geq 2\):\(\mu(D)\) is super-\(\kappa\) if and only if \(\delta(D) < 2\kappa(D)\).;\(\mu(D)\) is super-\(\lambda\) if and only if \(D \ncong \overrightarrow{K_2}\).