Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Light structures in infinite planar graphs without the strong isoperimetric property

Author: Bojan Mohar
Journal: Trans. Amer. Math. Soc. 354 (2002), 3059-3074
MSC (2000): Primary 05B45, 52B60, 52C20, 60J10
Published electronically: April 2, 2002
MathSciNet review: 1897390
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the discharging method can be successfully applied on infinite planar graphs of subexponential growth and even on those graphs that do not satisfy the strong edge isoperimetric inequality. The general outline of the method is presented and the following applications are given: Planar graphs with only finitely many vertices of degree $\le 5$ and with subexponential growth contain arbitrarily large finite submaps of the tessellation of the plane or of some tessellation of the cylinder by equilateral triangles. Every planar graph with isoperimetric number zero and with essential minimum degree $\ge3$ has infinitely many edges whose degree sum is at most 15. In particular, this holds for all graphs with minimum degree $\ge3$ and with subexponential growth. The cases without infinitely many edges whose degree sum is $\le14$ (or, similarly, $\le13$ or $\le 12$) are also considered. Several further results are obtained.

References [Enhancements On Off] (What's this?)

  • 1. A. Altshuler, Hamiltonian circuits in some maps on the torus, Discrete Math. 1 (1972) 299-314. MR 45:6651
  • 2. K. Appel, W. Haken, Every planar map is four colorable. Part I: Discharging, Ill. J. Math. 21 (1977) 429-490. MR 58:27598a
  • 3. O. Baues, N. Peyerimhoff, Curvature and geometry of tessellating plane graphs, Discrete Comput. Geom. 25 (2001) 141-159. MR 2001k:57004
  • 4. O. V. Borodin, A generalization of Kotzig's theorem and prescribed edge coloring of planar graphs, Mat. Zametki 48 (1990) no. 6, 22-28; English transl., Math. Notes 48 (1990), 1186-1190. MR 92e:05046
  • 5. A. Calogero, Strong isoperimetric inequality for the edge graph of a tiling of the plane, Arch. Math. (Basel) 61 (1993) 584-595. MR 94m:52024
  • 6. J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984) 787-794. MR 85m:58185
  • 7. I. Fabrici, S. Jendrol', Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs Combin. 13 (1997) 245-250. MR 98h:05112
  • 8. P. Gerl, Random walks on graphs with a strong isoperimetric inequality, J. Theoret. Probab. 1 (1988) 171-187. MR 89g:60216
  • 9. M. Gromov, Hyperbolic manifolds (according to Thurston and Jorgensen), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math. 842, Springer, Berlin-New York, 1981, pp. 40-53. MR 84b:53046
  • 10. M. Gromov, Hyperbolic groups. Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, pp. 75-263. MR 89e:20070
  • 11. B. Grünbaum, G. C. Shephard, Analogues for tilings of Kotzig's theorem on minimal weights of edges, in ``Theory and practice of combinatorics,'' North-Holland, Amsterdam-New York, 1982, pp. 129-140. MR 86k:52015
  • 12. N. Robertson, D. Sanders, P. Seymour, R. Thomas, The four-colour theorem, J. Combin. Theory Ser. B 70 (1997) 2-44. MR 98c:05065
  • 13. Zh.-X. He, O. Schramm, Hyperbolic and parabolic packings, Discrete Comput. Geom. 14 (1995) 123-149. MR 96h:52017
  • 14. O. Häggström, J. Jonasson, R. Lyons, Explicit isoperimetric constants, phase transitions in the random-cluster, and Bernoullicity, preprint, 2001.
  • 15. A. Kotzig, On the theory of Euler polyhedra (in Russian), Mat.-Fyz. Cas. Sloven. Akad. Vied 13 (1963) 20-31. MR 28:5375
  • 16. R. Lyons, Y. Peres, Probability on trees and networks, book manuscript, 2000.
  • 17. B. Mohar, Isoperimetric inequalities, growth, and the spectrum of graphs, Linear Algebra Appl. 103 (1988) 119-131. MR 89k:05071
  • 18. B. Mohar, Some relations between analytic and geometric properties of infinite graphs, Discrete Math. 95 (1991) 193-219. MR 93c:05113
  • 19. B. Mohar, Isoperimetric numbers and spectral radius of some infinite planar graphs, Math. Slovaca 42 (1992) 411-425. MR 94a:05003
  • 20. P. M. Soardi, Recurrence and transience of the edge graph of a tiling of the Euclidean plane, Math. Ann. 287 (1990) 613-626. MR 92b:52044
  • 21. T. Stehling, Über das Kotziggewicht normaler Pflasterungen, Resultate Math. 18 (1990) 347-354. MR 92a:52029
  • 22. W. Woess, A note on tilings and strong isoperimetric inequality, Math. Proc. Cambridge Philos. Soc. 124 (1998) 385-393. MR 99f:52026
  • 23. A. Zuk, On the norms of the random walks on planar graphs, Ann. Inst. Fourier (Grenoble) 47 (1997) 1463-1490. MR 99g:60127

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05B45, 52B60, 52C20, 60J10

Retrieve articles in all journals with MSC (2000): 05B45, 52B60, 52C20, 60J10

Additional Information

Bojan Mohar
Affiliation: Department of Mathematics, University of Ljubljana, 1111 Ljubljana, Slovenia

Received by editor(s): March 19, 2001
Published electronically: April 2, 2002
Additional Notes: Supported in part by the Ministry of Science and Technology of Slovenia, Research Project J1–0502–0101–00.
Article copyright: © Copyright 2002 American Mathematical Society