On semi-transitivity of (extended) Mycielski graphs

An orientation of a graph is semi-transitive if it is acyclic and shortcut-free. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalise several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. The Mycielski graph of an undirected graph is a larger graph, constructed in a certain way, that maintains the property of being triangle-free but enlarges the chromatic number. These graphs are important as they allow to prove the existence of triangle-free graphs with arbitrarily large chromatic number. An extended Mycielski graph is a certain natural extension of the notion of a Mycielski graph that we introduce in this paper. In this paper we characterise completely semi-transitive extended Mycielski graphs and Mycielski graphs of comparability graphs. We also conjecture a complete characterisation of semi-transitive Mycielski graphs. Our studies are a far-reaching extension of the result of Kitaev and Pyatkin on non-semi-transitive orientability of the Mycielski graph mu ( C 5 ) of the cycle graph C 5 . Using a recent result of Kitaev and Sun, we shorten the length of the original proof of non-semi-transitive orientability of mu ( C 5 ) from 2 pages to a few lines. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).