Packing degenerate graphs greedily

Abstract

We prove that if G is a family of graphs with at most n vertices each, with constant degeneracy, with maximum degree at most O (n/ log n), and with total number of edges at most, then G packs into the complete graph K n .

Almost all trees are almost graceful

“…We choose the J(v i ) to be uniformly distributed 2 over [ñ], and such that for most v i the intervals J(v i ) 2 A technical complication arises in that the extremes of the interval [ñ] are not covered as much as the rest; we will ignore this for now. and J(v j ), where v j is the parent of v i in the ordering, are equally far from and on opposite sides of 1 2 (1 + γ)n. In reality, we certainly do not choose vertex labels uniformly at random in the intervals; there is a great deal of dependency in the label choices. However, that dependency is sequential: conditioning on the labelling history of the first t − 1 vertices, we know the distribution of the t th vertex, and we will be able to show that the average of these distributions is close to uniform (in a sense which we will make precise later).…”

“…Because n is divisible by 2δ −4 0 γ −1 , and δ −1 0 is an integer, ℓ, m For each J ∈ J , we define the complementary interval J ∈ J to be the (unique) interval J such that the sum of the elements of J ∪ J is ℓ(ñ + 1). By definition, J and J are disjoint, one having only elements less than or equal to 1 2 ñ and the other having only elements greater than 1 2 ñ.…”

“…Conjecture 2 was wide open until recently when in [5,4] an approximate version for bounded degree trees was proven. 1 The main result of [5,4] was extended by Messuti, Rödl, and Schacht [17] to permit almost-spanning bounded degree graphs with limited expansion and later Ferber, Lee, and Mousset [8] allowed for spanning bounded degree graphs with limited expansion. Kim, Kühn, Osthus, and Tyomkyn [16] were able to obtain approximate packings of a complete graph with an arbitrary family of bounded degree graphs, and using this Joos, Kim, Kühn and Osthus [15] proved the Tree Packing Conjecture and Ringel's Conjecture exactly for arbitrary bounded degree trees.…”

“…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]. 1 The main result in [5,4] is more general and allows almost perfect packings of a complete graph with an arbitrary family of almost-spanning bounded degree trees, addressing also the Tree Packing Conjecture of Gyarfás.…”