Hostname: page-component-77f85d65b8-45ctf Total loading time: 0 Render date: 2026-04-19T13:22:02.228Z Has data issue: false hasContentIssue false
  • English
  • Français

Induced Forests in Regular Graphs with Large Girth

Published online by Cambridge University Press:  01 May 2008

CARLOS HOPPEN  and
NICHOLAS WORMALD
CARLOS HOPPEN
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, CanadaN2L 3G1 (e-mail: choppen@math.uwaterloo.ca, nwormald@uwaterloo.ca)
NICHOLAS WORMALD
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, CanadaN2L 3G1 (e-mail: choppen@math.uwaterloo.ca, nwormald@uwaterloo.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomized algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 3 , May 2008 , pp. 389 - 410
Copyright
Copyright © Cambridge University Press 2008

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.)

Article purchase

Temporarily unavailable

References

[1]Alon, N. and Spencer, J. (2000) The Probabilistic Method, Wiley.CrossRefGoogle Scholar
[2]Bau, S., Beineke, L. W., Du, G. M., Liu, Z. S. and Vandell, R. C. (2001) Decycling cubes and grids. Utilitas Math. 59 129–137.Google Scholar
[3]Bau, S., Wormald, N. C. and Zhou, S. (2002) Decycling number of random regular graphs. Random Struct. Alg. 21 397–413.CrossRefGoogle Scholar
[4]Beineke, L. W. and Vandell, R. C. (1997) Decycling graphs. J. Graph Theory 25 59–77.3.0.CO;2-H>CrossRefGoogle Scholar
[5]Bollobás, B. (1980) A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combin. 1 311–316.Google Scholar
[6]Duckworth, W. and Wormald, N. C. (2006) On the independent domination number of random regular graphs. Combin. Probab. Comput. 15 513–522.CrossRefGoogle Scholar
[7]Edwards, K. and Farr, G. (2001) Fragmentability of graphs. J. Combin. Theory Ser. B 82 30–37.CrossRefGoogle Scholar
[8]Erdős, P., Saks, M. and Sós, V. T. (1986) Maximum induced trees in graphs. J. Combin. Theory Ser. B 41 61–79.CrossRefGoogle Scholar
[9]Hurewicz, W. (1958) Lectures on Ordinary Differential Equations, MIT Press.CrossRefGoogle Scholar
[10]Karp, R. M. (1972) Reducibility among combinatorial problems. In Complexity of Computer Computations (Miller, R. E. and Thatcher, J. W., eds), Plenum, pp. 85–103.CrossRefGoogle Scholar
[11]Kirchhoff, G. (1847) Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72 497–508.CrossRefGoogle Scholar
[12]Lauer, J. and Wormald, N. C. (2007) Large independent sets in random graphs with large girth. J. Combin. Theory Ser. B 97 999–1009.CrossRefGoogle Scholar
[13]Li, D. M. and Liu, Y. P. (1999) A polynomial algorithm for finding the minimum feedback vertex set of a 3-regular simple graph. Acta Math. Sci. 19 375–381.CrossRefGoogle Scholar
[14]Liang, Y. D. (1994) On the feedback vertex problem in permutation graphs. Inform. Process. Lett. 52 123–129.CrossRefGoogle Scholar
[15]Liang, Y. D. and Chang, M. S. (1997) Minimum feedback vertex sets in cocomparability graphs and convex bipartite graphs. Acta Inform. 34 337–346.CrossRefGoogle Scholar
[16]Wormald, N. C. (1978) Some problems in the enumeration of labelled graphs. Doctoral thesis, Newcastle University.Google Scholar
[17]Wormald, N. C. (1981) The asymptotic distribution of short cycles in random regular graphs. J. Combin. Theory Ser. B 31 168–182.CrossRefGoogle Scholar
[18]Wormald, N. C. (1995) Differential equations for random processes and random graphs. Ann. Appl. Probab. 5 1217–1235.CrossRefGoogle Scholar