On semi-transitive orientability of triangle-free graphs Full article
| Journal |
Discussiones Mathematicae - Graph Theory
ISSN: 1234-3099 , E-ISSN: 2083-5892 |
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| Output data | Year: 2023, Volume: 43, Number: 2, Pages: 533-547 Pages count : 15 DOI: 10.7151/dmgt.2384 | ||||||
| Tags | semi-transitive orientation, triangle-free graph, Grötzsch graph, Mycielski graph, Chvátal graph, Toft's graph, circulant graph, Toeplitz graph | ||||||
| Authors |
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| Affiliations |
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Funding (1)
| 1 | Sobolev Institute of Mathematics | 0314-2019-0014 |
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 semitransitive 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 nonsemi- transitive graphs. In this paper we prove that the smallest trianglefree 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 trianglefree 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.
Cite:
Kitaev S.
, Pyatkin A.V.
On semi-transitive orientability of triangle-free graphs
Discussiones Mathematicae - Graph Theory. 2023. V.43. N2. P.533-547. DOI: 10.7151/dmgt.2384 WOS Scopus РИНЦ OpenAlex
On semi-transitive orientability of triangle-free graphs
Discussiones Mathematicae - Graph Theory. 2023. V.43. N2. P.533-547. DOI: 10.7151/dmgt.2384 WOS Scopus РИНЦ OpenAlex
Dates:
| Submitted: | Jul 13, 2020 |
| Accepted: | Nov 14, 2020 |
| Published online: | Dec 18, 2020 |
| Published print: | Mar 2, 2023 |
Identifiers:
| Web of science: | WOS:000737388800001 |
| Scopus: | 2-s2.0-85100710347 |
| Elibrary: | 46751967 |
| OpenAlex: | W3109076334 |