A note on Hameed’s conjecture on the semi-transitivity of Mycielski graphs

An orientation of a graph is semi-transitive if it contains no directed cycles and has no shortcuts. An undirected graph is semi-transitive if it can be oriented in a semi-transitive manner. The class of semi-transitive graphs includes several important graph classes. The Mycielski graph of an undirected graph is a larger graph constructed in a specific manner, which maintains the property of being triangle-free but increases the chromatic number.

In this note, we prove Hameed's conjecture, which states that the Mycielski graph of a graph $G$ is semi-transitive if and only if $G$ is a bipartite graph. Notably, our solution to the conjecture provides an alternative and shorter proof of the Hameed's result on a complete characterization of semi-transitive extended Mycielski graphs.