A short construction of highly chromatic digraphs without short cycles

Authors

  • Michael Severino University of Montana

DOI:

https://doi.org/10.11575/cdm.v9i2.62197

Keywords:

digraph, chromatic number, acyclic homomorphism, girth

Abstract

A natural digraph analogue of the graph-theoretic concept of an `independent set' is that of an `acyclic set', namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets. In the spirit of a famous theorem of P. Erd\H{o}s [Graph theory and probability, Canad. J. Math. {\bf11} (1959), 34--38], it was shown probabilistically in [D. Bokal et al., The circular chromatic number of a digraph, J. Graph Theory {\bf46} (2004), no. 3, 227--240] that there exist digraphs with arbitrarily large girth and chromatic number. Here we give a construction of such digraphs.

 

References

[1] J. Bang-Jensen and G. Gutin. Digraphs: Theory, Algorithms and Applications. Springer Mono- graphs in Mathematics. Springer-Verlag London Ltd., London, 2001.

[2] D. Bokal, G. Fijavˇz, M. Juvan, P. M. Kayll, and B. Mohar. The circular chromatic number of a digraph. J. Graph Theory, 46(3):227–240, 2004.

[3] P. Erd ̋os. Graph theory and probability. Canad. J. Math., 11:34–38, 1959.

[4] I. Kˇr ́ıˇz. A hypergraph-free construction of highly chromatic graphs without short cycles. Com-

binatorica, 9(2):227–229, 1989.

[5] L. Lov ́asz. On chromatic number of finite set-systems. Acta Math. Acad. Sci. Hungar., 19:59–

67, 1968.

[6] J. Neˇsetˇril and V. R ̈odl. A short proof of the existence of highly chromatic hypergraphs without

short cycles. J. Combin. Theory Ser. B, 27(2):225–227, 1979.

Downloads

Published

2014-12-30

Issue

Vol. 9 No. 2 (2014)

Section

Articles

License

 

This copyright statement was adapted from the statement for the University of Calgary Repository and from the statement for the Electronic Journal of Combinatorics (with permission).

The copyright policy for Contributions to Discrete Mathematics (CDM) is changed for all articles appearing in issues of the journal starting from Volume 15 Number 3.

Author(s) retain copyright over submissions published starting from  Volume 15 number 3.   When the author(s) indicate approval of the finalized version of the article provided by the technical editors of the journal and indicate approval, they grant to Contributions to Discrete Mathematics (CDM)  a world-wide, irrevocable, royalty free,  non-exclusive license as described below:

The author(s) grant to CDM the right to reproduce, translate (as defined below), and/or distribute the material, including the abstract, in print and electronic format, including but not limited to audio or video.

The author(s) agree that the journal may translate, without changing the content the material, to any medium or format for the purposes of preservation. 

The author(s) also agree that the journal may keep more than one copy of the article for the purposes of security, back-up, and preservation.

In granting the journal this license the author(s) warrant that the work is their original work and that they have the right to grant the rights contained in this license.

The authors represent that the work does not, to the best of their knowledge, infringe upon anyone’s copyright.

If the work contains material for which the author(s) do not hold copyright, the author(s) represent that the unrestricted permission of the copyright holder(s) to grant CDM the rights required by this license has been obtained, and that such third-party owned material is clearly identified and acknowledged within the text or content of the work.

The author(s) agree to ensure, to the extent reasonably possible, that further publication of the Work, with the same or substantially the same content, will acknowledge prior publication in CDM.

The journal highly recommends that the work be published with a Creative Commons license.   Unless otherwise arranged at the time the finalized version is approved and the licence granted with CDM,  the work will appear with the CC-BY-ND logo.  Here is the site to get more detail, and an excerpt from the site about the CC-BY-ND.  https://creativecommons.org/licenses/

Attribution-NoDerivs
CC BY-ND

This license lets others reuse the work for any purpose, including commercially; however, it cannot be shared with others in adapted form, and credit must be provided to you.