An approximate version of the tree packing conjecture

We prove that for any pair of constants ɛ > 0 and Δ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most Δ, and with at most (n2) edges in total packs into $${K_{(1 + \varepsilon )n}}$$. This implies asymptotic versions of the Tree Packing Conjecture of Gyárfás from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

[1]  W. Marsden.  I and J , 2012 .

[2]  Yair Caro,et al.  A note on packing trees into complete bipartite graphs and on fishburn's conjecture , 1990, Discret. Math..

[3]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[4]  T. Bohman.  The triangle-free process , 2008, 0806.4375.

[5]  Fan Chung Graham,et al.  Quasi-random graphs , 1988, Comb..

[6]  Béla Bollobás,et al.  Some remarks on packing trees , 1983, Discret. Math..

[7]  Alan M. Frieze,et al.  Packing Tree Factors in Random and Pseudo-random Graphs , 2014, Electron. J. Comb..

[8]  Raphael Yuster.  On packing trees into complete bipartite graphs , 1997, Discret. Math..

[9]  Edward Dobson,et al.  Packing Trees into the Complete Graph , 2002, Combinatorics, Probability and Computing.

[10]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[11]  Ben Barber,et al.  Random Walks on Quasirandom Graphs , 2013, Electron. J. Comb..

[12]  Alon Itai,et al.  Packing Trees , 1995, ESA.

[13]  Stephen Suen,et al.  A correlation inequality and a Poisson limit theorem for nonoverlapping balanced subgraphs of a random graph , 1990, Random Struct. Algorithms.

[14]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[15]  Paul A. Catlin,et al.  Subgraphs of graphs, I , 1974, Discret. Math..

[16]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[17]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

[18]  Jeong Han Kim,et al.  The Ramsey Number R(3, t) Has Order of Magnitude t2/log t , 1995, Random Struct. Algorithms.

[19]  A. Thomason.  Pseudo-Random Graphs , 1987 .

[20]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[21]  Arthur M. Hobbs,et al.  Packing trees in complete graphs , 1987, Discret. Math..

[22]  Peter C. Fishburn.  Balanced Integer Arrays: A Matrix Packing Theorem , 1983, J. Comb. Theory, Ser. A.

[23]  Béla Bollobás,et al.  Packings of graphs and applications to computational complexity , 1978, J. Comb. Theory, Ser. B.

[24]  Bruce A. Reed,et al.  Concentration for self‐bounding functions and an inequality of Talagrand , 2006, Random Struct. Algorithms.

[25]  N. Alon,et al.  The Probabilistic Method: Alon/Probabilistic , 2008 .

[26]  József Balogh,et al.  On the Tree Packing Conjecture , 2012, SIAM J. Discret. Math..

[27]  Raphael Yuster,et al.  Packing Graphs: The packing problem solved , 1996, Electron. J. Comb..