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Title: C–H Functionalization Approach for the Synthesis of Chiral C 2 -Symmetric 1,5-Cyclooctadiene Ligands
Award ID(s):
1700982
PAR ID:
10143949
Author(s) / Creator(s):
; ; ;
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
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  2. Abstract

    For a subgraph$G$of the blow-up of a graph$F$, we let$\delta ^*(G)$be the smallest minimum degree over all of the bipartite subgraphs of$G$induced by pairs of parts that correspond to edges of$F$. Johansson proved that if$G$is a spanning subgraph of the blow-up of$C_3$with parts of size$n$and$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$, then$G$contains$n$vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$G$is a spanning subgraph of the blow-up of$C_k$with parts of size$n$and$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$, then$G$contains$n$vertex disjoint copies of$C_k$such that each$C_k$intersects each of the$k$parts exactly once. A similar conjecture was also made by Fischer and the case$k=3$was proved for large$n$by Magyar and Martin.

    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$G$to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.

     
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