Abstract
Given D and γ>0, whenever c>0 is sufficiently small and n sufficiently large, if G is a family of D-degenerate graphs of individual orders at most n, maximum degrees at most cnlogn, and total number of edges at most (1−γ)(n2), then G packs into the complete graph Kn. Our proof proceeds by analysing a natural random greedy packing algorithm.
Highlights
A packing of a family G = {G1, . . . , Gk} of graphs into a graph H is a colouring of the edges of H with the colours 0, 1, . . . , k such that the edges of colour i form an isomorphic copy ofGi for each 1 ≤ i ≤ k
Packing problems have been studied in graph theory for several decades
Many classical theorems and conjectures of extremal graph theory can be written as packing problems
Summary
Note that without restriction the problem of packing a given G into a given H is NP-complete (the survey [27] gives several NP-completeness results of which the one in [9] is arguably the most convincing), so in particular we do not expect to find any finite list of simple obstructions to the general packing problem It follows that ‘dense’ in the meta-conjecture cannot mean large edge-density: one can artificially boost edge density without changing the outcome of this decision problem by taking the disjoint union with a very large clique and adding large connected graphs to G which perfectly pack the very large clique.
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