The minimum degree threshold for perfect graph packings

The minimum degree threshold for perfect graph packings

Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing. More precisely, let δ(H, n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-packing. We show that

Minimum degree threshold for bipartite graph tiling

We answer a question of Zhao [SIAM J. Disc. Math. 23 vol.2, (2009), 888-900] that determines the minimum degree threshold for a bipartite graph G to contain an H-factor (a perfect tiling of G with H) for any bipartite graph H. We also show that this threshold is best possible up to a constant depending only on H. This result can be viewed as an analog to Kuhn and Osthus' result [Combinatorica 2...

Minimum degree thresholds for bipartite graph tiling

Given a bipartite graph H and a positive integer n such that v(H) divides 2n, we define the minimum degree threshold for bipartite H-tiling, δ2(n,H), as the smallest integer k such that every bipartite graph G with n vertices in each partition and minimum degree δ(G) ≥ k contains a spanning subgraph consisting of vertex-disjoint copies of H. Zhao, Hladký-Schacht, Czygrinow-DeBiasio determined δ...

Graph Minors and Minimum Degree

Let Dk be the class of graphs for which every minor has minimum degree at most k. Then Dk is closed under taking minors. By the Robertson-Seymour graph minor theorem, Dk is characterised by a finite family of minor-minimal forbidden graphs, which we denote by D̂k. This paper discusses D̂k and related topics. We obtain four main results: 1. We prove that every (k + 1)-regular graph with less than ...

Graph packings and the degree prescribed subgraph problem