DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 22(1) (2002) 123-148
DOI: 10.7151/dmgt.1163

CONDITIONS FOR β-PERFECTNESS

 Judith Keijsper

University of Twente
Faculty of Mathematical Sciences
7500 AE Enschede, The Netherlands

 Meike Tewes

Freiberg University
Faculty of Mathematics and Computer Sciences
09596 Freiberg, Germany

Abstract

A β-perfect graph is a simple graph G such that χ(G′) = β(G′) for every induced subgraph G′ of G, where χ(G′) is the chromatic number of G′, and β(G′) is defined as the maximum over all induced subgraphs H of G′ of the minimum vertex degree in H plus 1 (i.e., δ(H)+1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily).

The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden induced subgraphs, for a graph to be β-perfect. We give new sufficient conditions and make improvements to sufficient conditions previously given by others. We also mention a necessary condition which generalizes the fact that no β-perfect graph contains an even hole.

Keywords: chromatic number, colouring number, polynomial time.

2000 Mathematics Subject Classification: 05C15, 05C75.

References

[1] L.W. Beineke, Characterizations of derived graphs, J. Combin. Theory 9 (1970) 129-135, doi: 10.1016/S0021-9800(70)80019-9.
[2] R.L. Brooks, On colouring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941) 194-197, doi: 10.1017/S030500410002168X.
[3] M. Conforti, G. Cornuéjols, A. Kapoor and K. Vusković, Finding an even hole in a graph, in: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (1997) 480-485, doi: 10.1109/SFCS.1997.646136.
[4] G.A. Dirac, On rigid circuit graphs, Abh. Math. Univ. Hamburg 25 (1961) 71-76, doi: 10.1007/BF02992776.
[5] P. Erdős and A. Hajnal, On the chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hungar. 17 (1966) 61-99, doi: 10.1007/BF02020444.
[6] C. Figueiredo and K. Vusković, A class of β-perfect graphs, Discrete Math. 216 (2000) 169-193, doi: 10.1016/S0012-365X(99)00240-X.
[7] H.-J. Finck and H. Sachs, Über eine von H.S. Wilf angegebene Schranke für die chromatische Zahl endlicher Graphen, Math. Nachr. 39 (1969) 373-386, doi: 10.1002/mana.19690390415.
[8] T.R. Jensen and B. Toft, Graph colouring problems (Wiley, New York, 1995).
[9] S.E. Markossian, G.S. Gasparian and B.A. Reed, β-perfect graphs, J. Combin. Theory (B) 67 (1996) 1-11, doi: 10.1006/jctb.1996.0030.

Received 10 August 2000
Revised 3 July 2001