Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T00:58:25.192Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost Every Tree With m Edges Decomposes K2m,2m

Published online by Cambridge University Press:  03 December 2013

M. DRMOTA  and
A. LLADÓ
M. DRMOTA
Affiliation:
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Vienna, Austria (e-mail: [email protected]
A. LLADÓ
Affiliation:
Departament de Matemàtica Aplicada~IV, Universitat Politècnica de Catalunya–BarcelonaTech, Barcelona, Spain (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K2m,2m, a result connected to a conjecture of Graham and Häggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a random tree have roughly equal size.

Keywords

Primary 05C51Secondary 05C80
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 1 , January 2014 , pp. 50 - 65
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]Alon, N. (1999) Combinatorial Nullstellensatz. Combin. Probab. Comput. 8 7–29.CrossRefGoogle Scholar
[2]Alon, N. (2000) Additive Latin transversals. Israel J. Math. 117 125–130.Google Scholar
[3]Càmara, M., Lladó, A. and Moragas, J. (2009) On a conjecture of Graham and Häggkvist with the polynomial method. Europ. J. Combin. 30 1585–1592.Google Scholar
[4]Dasgupta, S., Károlyi, G., Serra, O. and Szegedy, B. (2001) Transversals of additive Latin squares. Israel J. Math. 126 17–28.Google Scholar
[5]Drmota, M. (2009) Random Trees, Springer.CrossRefGoogle Scholar
[6]Drmota, M. and Gittenberger, B. (1999) The distribution of nodes of given degree in random trees. J. Graph Theory 31 227–253.3.0.CO;2-6>CrossRefGoogle Scholar
[7]Drmota, M. and Gittenberger, B. (2010) The shape of unlabeled rooted random trees. Europ. J. Combin. 31 2028–2063.CrossRefGoogle Scholar
[8]Flajolet, P. and Odlyzko, A. (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216–240.CrossRefGoogle Scholar
[9]Gallian, J. A. (2007) A dynamic survey of graph labeling. Electron. J. Combin. 5 #DS6.Google Scholar
[10]Häggkvist, R. L. (1989) Decompositions of complete bipartite graphs. In Surveys in Combinatorics (Siemons, J., ed.), Cambridge University Press, pp. 115–146.Google Scholar
[11]Hwang, H. K. (1994) Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques. PhD thesis, Ecole Polytechnique, New York.Google Scholar
[12]Kézdy, A. E. (2006) ρ-valuations for some stunted trees. Discrete Math. 306 2786–2789.Google Scholar
[13]Kézdy, A. E. and Snevily, H. S. (2002) Distinct sums modulo n and tree embeddings. Combin. Probab. Comput. 11 35–42.CrossRefGoogle Scholar
[14]Lladó, A. and López, S. C. (2005) Edge-decompositions of Kn,n into isomorphic copies of a given tree. J. Graph Theory 48 1–18.Google Scholar
[15]Lladó, A., López, S. C. and Moragas, J. (2010) Every tree is a large subtree of a tree that decomposes Kn or Kn,n. Discrete Math. 310 838–842.Google Scholar
[16]Otter, R. (1948) The number of trees. Ann. of Math. (2) 49 583–599.Google Scholar
[17]Ringel, G. (1964) Problem 25. In Theory of Graphs and its Applications, Nakl. CSAV, Praha, p. 162.Google Scholar
[18]Robinson, R. W. and Schwenk, A. J. (1975) The distribution of degrees in a large random tree. Discrete Math. 12 359–372.CrossRefGoogle Scholar
[19]Snevily, H. (1997) New families of graphs that have α-labelings. Discrete Math. 170 185–194.CrossRefGoogle Scholar
[20]Yuster, R. (2000) Packing and decomposition of graphs with trees. J. Combin. Theory Ser. B 78 123–140.CrossRefGoogle Scholar