# On semi-transitive orientability of triangle-free graphs

Kitaev, Sergey and Pyatkin, Artem
(2020)
*On semi-transitive orientability of triangle-free graphs.*
Discussiones Mathematicae Graph Theory.
ISSN 1234-3099
(In Press)

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## Abstract

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path v0→v1→⋯→vk either there is no arc between v0 and vk, or vi→vj is an arc for all 0≤i<j≤k. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence of non-semi-transitive triangle-free graphs was established via Erdős' theorem by Halldórsson and the authors in 2011. However, no explicit examples of such graphs were known until recent work of the first author and Saito who have shown computationally that a certain subgraph on 16 vertices of the triangle-free Kneser graph K(8,3) is not semi-transitive, and have raised the question on the existence of smaller triangle-free non-semi-transitive graphs. In this paper we prove that the smallest triangle-free 4-chromatic graph on 11 vertices (the Grötzsch graph) and the smallest triangle-free 4-chromatic 4-regular graph on 12 vertices (the Chvátal graph) are not semi-transitive. Hence, the Grötzsch graph is the smallest triangle-free non-semi-transitive graph. We also prove the existence of semi-transitive graphs of girth 4 with chromatic number 4 including a small one (the circulant graph C(13;1,5) on 13 vertices) and dense ones (Toft's graphs). Finally, we show that each 4-regular circulant graph (possibly containing triangles) is semi-transitive.

#### ORCID iDs

Kitaev, Sergey ORCID: https://orcid.org/0000-0003-3324-1647 and Pyatkin, Artem;Item type: Article ID code: 74710 Dates: DateEvent14 November 2020Published14 November 2020AcceptedKeywords: semi-transitive orientation, triangle-free graph, Grotzsch graph, Mycielski graph, Chvatal graph, Toft's graph, circulant graph, Toeplitz graph, Mathematics, Applied Mathematics, Discrete Mathematics and Combinatorics Subjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 26 Nov 2020 11:55 Last modified: 28 Apr 2021 00:40 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/74710