 NSFPAR ID:
 10063152
 Publisher / Repository:
 American Institute of Physics
 Date Published:
 Journal Name:
 Journal of Physical and Chemical Reference Data
 Volume:
 47
 Issue:
 3
 ISSN:
 00472689
 Page Range / eLocation ID:
 033101
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract For a subgraph
of the blowup 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 blowup 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 blowup 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.$G$ 
null (Ed.)The water reactivity of the boroauride complex ([Au(B 2 P 2 )][K(18c6)]; (B 2 P 2 , 9,10bis(2(diisopropylphosphino)phenyl)9,10dihydroboranthrene) and its corresponding twoelectron oxidized complex, Au(B 2 P 2 )Cl, are presented. Au(B 2 P 2 )Cl is tolerant to H 2 O and forms the hydroxide complex Au(B 2 P 2 )OH in the presence of H 2 O and triethylamine. [Au(B 2 P 2 )]Cl and [Au(B 2 P 2 )]OH are poor Lewis acids as judged by the Gutmann–Becket method, with [Au(B 2 P 2 )]OH displaying facile hydroxide exchange between B atoms of the DBA ring as evidenced by variable temperature NMR spectroscopy. The reduced boroauride complex [Au(B 2 P 2 )] − reacts with 1 equivalent of H 2 O to produce a hydride/hydroxide product, [Au(B 2 P 2 )(H)(OH)] − , that rapidly evolves H 2 upon further H 2 O reaction to yield the dihydroxide compound, [Au(B 2 P 2 )(OH) 2 ] − . [Au(B 2 P 2 )]Cl can be regenerated from [Au(B 2 P 2 )(OH) 2 ] − via HCl·Et 2 O, providing a synthetic cycle for H 2 evolution from H 2 O enabled by O–H oxidative addition at a diboraanthracene unit.more » « less