Packing minor closed families of graphs

Motivated by a conjecture of Gyárfás, recently Böttcher, Hladký, Piguet, and Taraz showed that every collection T1, . . . , Tn of trees on n vertices with ∑n i=1 e(Ti) ≤ ( n 2 ) and with bounded maximum degree, can be packed into the complete graph on (1 + o(1))n vertices. We generalize this result where we relax the restriction of packing families of trees to families of graphs of any given non-trivial minor closed class of graphs. A packing of a sequence of graphs F = (F1, . . . , Fn) into a graph H is a collection of edge-disjoint subgraphs H1, . . . ,Hn ⊆ H, such that Hi is isomorphic to Fi for every i ∈ [n]. A well-known conjecture of Gyárfás [4] states that a sequence of trees T = (T1, . . . , Tn), where v(Ti) = i for every i ∈ [n], packs into Kn. Note that the sum of the edges over T is precisely ( n 2 ) , hence the packing of T should use all the edges of Kn. A restricted approximate version where the host graph is a clique on (1+o(1))n vertices and the trees have at most n vertices, bounded maximum degree, and the sum over all edges is at most ( n 2 ) , was proved by Böttcher, Hladký, Piguet, and Taraz [3]. We extend this result to sequences of graphs from any non-trivial minor closed family. Theorem 1. For any ε > 0, ∆ ∈ N, and any non-trivial minor-closed family G there exists n0 ∈ N such that for every n ≥ n0 the following holds. If F = (F1, . . . , Fn) is a sequence of graphs from G, each having order at most n and maximum degree at most ∆, such that ∑n i=1 e(Fi) ≤ ( n 2 ) , then F packs into K(1+ε)n. The main idea in the proof is to remove a small separator from each graph, in such a way that all components have small constant size, then pack the components into a large clique contained into K(1+ε)n and use the remaining vertices for the separators. In fact we prove a more general result and derive Theorem 1 from that, and the Separator Theorem of Alon, Seymour, and Thomas [1]. We define a (δ, s)-separable family as a set of graphs each having the property that by removing δ proportion of the vertices, the resulting components have size at most s. Theorem 2. For any ε > 0 and ∆ ∈ N there exists δ > 0 such that for every s ∈ N and any (δ, s)-separable family G there exists n0 ∈ N such that for every n ≥ n0 the following holds. If F = (F1, . . . , Fn) is a sequence of graphs from G each having order at most n and maximum degree at most ∆, such that ∑n i=1 e(Fi) ≤ ( n 2 ) , then F packs into K(1+ε)n. As mentioned above, Theorem 1 easily follows from Theorem 2. In fact, the Separator Theorem states that for any minor closed family G there exists a constant c such that a graph G ∈ G of order n has a separator of order at most cn1/2 and the components have size at most n/2. Applying such result recursively to each component, for i > 0 iterations, leads to a separator U ⊆ V (G) such that