TY - JOUR

T1 - On Semi-Transitive Orientability of Triangle-Free Graphs

AU - Kitaev, Sergey

AU - Pyatkin, Artem

N1 - Publisher Copyright: © 2020 Sergey Kitaev et al., published by Sciendo. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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 Erdos' 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.

AB - 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 Erdos' 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.

KW - Chvátal graph

KW - circulant graph

KW - Grötzsch graph

KW - Mycielski graph

KW - semi-transitive orientation

KW - Toeplitz graph

KW - Toft's graph

KW - triangle-free graph

UR - http://www.scopus.com/inward/record.url?scp=85100710347&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/3347768a-1118-3477-a305-74d31a06d1b9/

U2 - 10.7151/dmgt.2384

DO - 10.7151/dmgt.2384

M3 - Article

AN - SCOPUS:85100710347

JO - Discussiones Mathematicae - Graph Theory

JF - Discussiones Mathematicae - Graph Theory

SN - 1234-3099

ER -