A bandwidth theorem for approximate decompositions

Abstract

We provide a degree condition on a regular n‐vertex graph G which ensures the existence of a near optimal packing of any family scriptH of bounded degree n‐vertex k‐chromatic separable graphs into G. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of Böttcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let δk be the infimum over all δ⩾1/2 ensuring an approximate Kk‐decomposition of any sufficiently large regular n‐vertex graph G of degree at least δn. Now suppose that G is an n‐vertex graph which is close to r‐regular for some r⩾(δk+ofalse(1false))n and suppose that H1,⋯,Ht is a sequence of bounded degree n‐vertex k‐chromatic separable graphs with 0true∑ie(Hi)⩽(1−ofalse(1false))e(G). We show that there is an edge‐disjoint packing of H1,⋯,Ht into G. If the Hi are bipartite, then r⩾(1/2+o(1))n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs G of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.

The bandwidth theorem for locally dense graphs

“…Since the bandwidth theorem was proven, a number of variants of the result have been obtained, including for arrangeable graphs [10] and degenerate graphs [30] and in the setting of random and pseudorandom graphs [1,5,23], as well as for robustly expanding graphs [24]. Very recently, a bandwidth theorem for approximate decompositions was proven by Condon, Kim, Kühn, and Osthus [12], whilst Glock and Joos [20] proved a -bounded edge colouring extension of Theorem 1.1. A general embedding result of Böttcher, Montgomery, Parczyk, and Person [6] also implies a bandwidth theorem in the setting of randomly perturbed graphs.…”

Optimal packings of bounded degree trees

“…Another direction is to consider host graphs of large degree. In particular, a very recent result in [9] implies that an approximate version of the tree packing conjecture for bounded degree trees holds for regular host graphs of degree r ≥ (1/2 + o(1))n. (In fact, the results in [9] extend to approximate decompositions of almost regular n-vertex host graphs of large degree into arbitrary n-vertex separable bounded degree graphs. )…”

Resolution of the Oberwolfach problem

“…In particular, Allen, Böttcher, Hladký, and Piguet [2] considered approximate decompositions into graphs of bounded degeneracy and maximum degree o(n/ log n) whenever the host graph G is sufficiently quasirandom, and Kim, Kühn, Osthus, and Tyomkyn [29] considered approximate decompositions into graphs of bounded degree in host graphs G satisfying weaker quasirandom properties (namely, ε-superregularity, see Section 3.3). Their resulting blow-up lemma for approximate decompositions was a key ingredient for [12,25] (and thus for the current paper too). It also implies that an approximate solution to the Oberwolfach problem can always be found (this was obtained independently by Ferber, Lee, and Mousset [17]).…”

“…In this paper, at a very high level, we also pursue such an approach. As approximate decomposition results, we exploit a hypergraph matching argument due to Alon and Yuster [3] (which in turn is based on the Rödl nibble via the Pippenger-Spencer theorem [36]) and a bandwidth theorem for approximate decompositions due to Condon, Kim, Kühn, and Osthus [12]. Our absorption procedure utilizes as a key element a very special case of a recent result of Keevash on resolvable designs [27].…”

“…This paper is organised as follows. In Section 2 we provide an overview of our approach and in Section 3 we collect several embedding and decomposition results including the bandwidth theorem for approximate decompositions from [12] and a special case of a result of Keevash [27] on resolvable designs. We prove Theorem 1.3 in Section 4 and add some concluding remarks in Section 5.…”