On semi-transitive orientability of triangle-free graphs

Kitaev, Sergey  ; Pyatkin, Artem V.

Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 533-547

Résumé

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path v_0→ v_1→⋯→ v_k either there is no arc between v_0 and v_k, or v_i→ v_j is an arc for all 0≤ i lt;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.

Détail

Keywords: semi-transitive orientation, triangle-free graph, Gr\"otzsch graph, Mycielski graph, Chv\'atal graph, Toft's graph, circulant graph, Toeplitz graph