Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T08:51:35.645Z Has data issue: false hasContentIssue false
  • English
  • Français

On Independent Sets in Graphs with Given Minimum Degree

Published online by Cambridge University Press:  16 September 2013

HIU-FAI LAW  and
COLIN McDIARMID
HIU-FAI LAW
Affiliation:
Mathematisches Seminar der Universität Hamburg, Germany (e-mail: hiufai.law@gmail.com)
COLIN McDIARMID
Affiliation:
Department of Statistics, Oxford University, UK (e-mail: cmcd@stats.ox.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider numbers and sizes of independent sets in graphs with minimum degree at least d. In particular, we investigate which of these graphs yield the maximum numbers of independent sets of different sizes, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.

Keywords

Primary 05C35Secondary 05C6905C07
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 6 , November 2013 , pp. 874 - 884
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alexander, J., Cutler, J. and Mink, T. (2012) Independent sets in graphs with given minimum degree. Electron. J. Combin. 19 #37.Google Scholar
[2]Carroll, T., Galvin, D. and Tetali, P. (2009) Matchings and independent sets of a fixed size in regular graphs. J. Combin. Theory Ser. A 116 1219–1227.Google Scholar
[3]Cutler, J. and Radcliffe, A. J. (2011) Extremal problems for independent set enumeration. Electron. J. Combin. 18 #169.Google Scholar
[4]Engbers, J. and Galvin, D. (2013) Counting independent sets of a fixed size in graphs with a minimum degree. J. Graph Theory DOI 10.1002/jgt.21756Google Scholar
[5]Galvin, D. (2011) Two problems on independent sets in graphs. Discrete Math. 311 2105–2112.CrossRefGoogle Scholar
[6]Gutman, I. and Harary, F. (1983) Generalizations of the matching polynomial. Utilitas Math. 24 97–106.Google Scholar
[7]Kahn, J. (2001) An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput. 10 219–237.Google Scholar
[8]Lin, S. B. and Lin, C. (1995) Trees and forests with large and small independent indices. Chinese J. Math. 23 199–210.Google Scholar
[9]McDiarmid, C., Steger, A. and Welsh, D. (2005) Random planar graphs. J. Combin. Theory Ser. B 93 187–205.CrossRefGoogle Scholar
[10]Prodinger, H. and Tichy, R. F. (1982) Fibonacci numbers of graphs. Fibonacci Quart. 20 16–21.Google Scholar
[11]Scott, A. D. and Sokal, A. D. (2005) The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Statist. Phys. 118 1151–1261.Google Scholar
[12]Zhao, Y. (2010) The number of independent sets in a regular graph. Combin. Probab. Comput. 19 315–320.CrossRefGoogle Scholar