Packing spanning graphs from separable families

Abstract

Let G be a separable family of graphs. Then for all positive constants ǫ and ∆ and for every sufficiently large integer n, every sequence G 1 , . . . , G t ∈ G of graphs of order n and maximum degree at most ∆ such that e(G 1 )+· · ·+e(G t ) ≤ (1−ǫ) n 2 packs into K n . This improves results of Böttcher, Hladký, Piguet, and Taraz when G is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás and Lehel (1976) for the case that all trees have maximum degree at most ∆.The proof uses the local resilience of random graphs and a special multi-stage packing procedure.

Packing trees of unbounded degrees in random graphs

“…Finally, as the event D N holds with probability at least 1 − 3n −8 , see (10), the second assertion of Theorem 1.6 follows.…”

“…Generalising this result, Messuti, Rödl, and Schacht [18] proved that the same conclusion holds under the weaker assumption that all the T i belong to some fixed minor-closed family. Recently, Ferber, Lee, and Mousset [10] improved this result by showing that these graphs can be packed into K n . Even more recently, Kim, Kühn, Osthus, and Tyomkyn [16] extended the result of [10] to arbitrary graphs with bounded maximum degree.…”

“…Recently, Ferber, Lee, and Mousset [10] improved this result by showing that these graphs can be packed into K n . Even more recently, Kim, Kühn, Osthus, and Tyomkyn [16] extended the result of [10] to arbitrary graphs with bounded maximum degree. These developments imply the following approximate versions of Conjectures 1.1 and 1.2.…”

Almost all trees are almost graceful

“…However the methods used in these papers do not seem not to hint at any approaches for graceful tree labellings. After this paper was made public, Allen, Böttcher, Hladký, and Piguet [2, 1] showed an almost perfect packing result for any family of possibly spanning graphs of with maximum degree cn/ log n and constant degeneracy, thus improving (in the setting of complete host graphs) upon [5,17,8,16].…”

Packing degenerate graphs greedily

“…The paper [6] shows that one can replace trees with graphs from any nontrivial minor-closed family. This was improved in [2] by allowing the graphs to be packed to be spanning. The paper [5] proves a near-perfect packing result for families of graphs with bounded maximum degree which are otherwise unrestricted.…”