- Award ID(s):
- 1700982
- PAR ID:
- 10143949
- Date Published:
- Journal Name:
- Organic Letters
- Volume:
- 21
- Issue:
- 24
- ISSN:
- 1523-7060
- Page Range / eLocation ID:
- 9864 to 9868
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
null (Ed.)
-
Abstract For a subgraph
of the blow-up of a graph$G$ , we let$F$ be the smallest minimum degree over all of the bipartite subgraphs of$\delta ^*(G)$ induced by pairs of parts that correspond to edges of$G$ . Johansson proved that if$F$ is a spanning subgraph of the blow-up of$G$ with parts of size$C_3$ and$n$ , then$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$ contains$G$ vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$n$ is a spanning subgraph of the blow-up of$G$ with parts of size$C_k$ and$n$ , then$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$ contains$G$ vertex disjoint copies of$n$ such that each$C_k$ intersects each of the$C_k$ parts exactly once. A similar conjecture was also made by Fischer and the case$k$ was proved for large$k=3$ by Magyar and Martin.$n$ In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of
to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.$G$