Random Planar Graphs With Given Average Degree

Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Volumn , Issue , 2007, Pages

  Gerke, Stefanie ^a   McDiarmid, Colin ^b   Steger, Angelika ^a   Weißl, Andreas ^a  

^a ETH ZURICH   (Switzerland)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Automorphisms; Growth constants; Planar graphs; Unlabelled graphs

Indexed keywords

EID: 84920752228     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1093/acprof:oso/9780198571278.003.0006     Document Type: Chapter

References (22)

1. E. A. Bender, Z. C. Gao, and N. C. Wormald, The number of labeled 2- connected planar graphs, Electronic Journal of Combinatorics 9 (2002) #R43.
2. M. Bodirsky, C. Gröpl, and M. Kang, Generating labeled planar graphs uniformly at random, Springer LNCS (ICALP 2003) 2719 (2003) 1095-1107.
3. N. Bonichon, C. Gavoille, and N. Hanusse, An information-theoretic upper bound of planar graphs using triangulations, Springer LNCS (STACS 2005) 2608 (2003) 499-510.
4. A. Denise, M. Vasconcellos, and D. Welsh, The random planar graph, Congressus numerantium 113 (1996) 61-79.
5. C. Dowden, Private communication (2005).
6. E. Fusy, Quadratic exact-size and linear approximate-size random generation of planar graphs, DMTCS Proceedings AD (2005) pp. 125-138.
7. E. Fusy, D. Poulalhon, and G. Schaeffer, Dissections and trees, with applications to optimal mesh encoding and to random sampling, Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA 2005) (2005) pp. 690-699.
8. S. Gerke and C. McDiarmid, On the number of edges in a random planar graph, Combinatorics, Probability, and Computing 13 (2004) 165-183.
9. S. Gerke, C. McDiarmid, A. Steger, and A. Weißl, Random planar graphs with n nodes and a fixed number of edges, Proceedings of the 16th ACMSIAM Symposium on Discrete Algorithms (SODA 2005) (2005) pp. 999-1007.
10. O. Giménez and M. Noy, Estimating the growth constant of labelled planar graphs, Proceedings of the 3rd Colloqium on Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities (2004) pp. 133-139.
11. O. Giménez and M. Noy, Asymptotic enumeration and limit laws of planar graphs (2005) arXiv:math.CO/0501269.
12. R. J. Lipton and R. E. Tarjan, A separator theorem for planar graphs, SIAM J. Appl. Math. 36/2 (1979) 177-189.
13. C. McDiarmid, A. Steger, and D. Welsh, Random graphs from planar and other addable classes, Topics in Discrete Mathematics (M. Klazar, (p.102) J. Kratochvil, M. Loebl, J. Matoušek, R. Thomas, P. Valtr, eds) (2006) 231-246.
14. C. McDiarmid, A. Steger, and D. Welsh, Random planar graphs, J. Combin. Theory Ser. B 93 (2005) 187-205.
15. M. Noy, Random planar graphs and the number of planar graphs (2005), this volume.
16. D. Osthus, H. J. Prömel, and A. Taraz, On random planar graphs, the number of planar graphs and their triangulations, J. Combin. Theory Ser. B 88 (2003) 119-134.
17. C. Papadimitriou and M. Yannakakis, The clique problem for planar graphs, Information Processsing Letters 13 (1981) 131-133.
18. L. B. Richmond and N. C. Wormald, Random triangulations of the plane, Eur. J. Combin. 9 (1988) 61-71.
19. N. Robertson, D. Sanders, P. D. Seymour, and R. Thomas, The fourcolour theorem, J. Combin. Theory Ser. B 70 (1997) 2-44.
20. W. T. Tutte, A census of planar triangulations, Canadian J. Math. 14 (1962) 21-38.
21. J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge (1992).
22. H. Whitney, Congruent Graphs and the Connectivity of Graphs, Amer. J. Math. 54/1 (1932) 150-168.