A note on semi-transitivity of Mycielski graphs

TL;DR

This paper proves Hameed's conjecture that the Mycielski graph of a graph is semi-transitive if and only if the original graph is bipartite, providing a shorter proof of a key characterization in graph theory.

Contribution

It confirms Hameed's conjecture and offers an alternative, concise proof of the complete characterization of semi-transitive extended Mycielski graphs.

Findings

Mycielski graph of a bipartite graph is semi-transitive

Mycielski graph of a non-bipartite graph is not semi-transitive

Shorter proof of the characterization of semi-transitive extended Mycielski graphs

Abstract

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.