Finding any given 2‐factor in sparse pseudorandom graphs efficiently

Given an $n$‐vertex pseudorandom graph $G$ and an $n$‐vertex graph $H$ with maximum degree at most two, we wish to find a copy of $H$ in $G$, that is, an embedding $\phi :V\left(H\right)\to V\left(G\right)$ so that $\phi \left(u\right)\phi \left(v\right)\in E\left(G\right)$ for all $uv\in E\left(H\right)$. Particular instances of this problem include finding a triangle‐factor and finding a Hamilton cycle in $G$. Here, we provide a deterministic polynomial time algorithm that finds a given $H$ in any suitably pseudorandom graph $G$. The pseudorandom graphs we consider are $\left(p,\lambda \right)$‐bijumbled graphs of minimum degree which is a constant proportion of the average degree, that is, $\mathrm{\Omega }\left(pn\right)$. A $\left(p,\lambda \right)$‐bijumbled graph is characterised through the discrepancy property: $|e\left(A,B\right)?p|A||B||<\lambda \sqrt{|A||B|}$ for any two sets of vertices $A$ and $B$. Our condition $\lambda =O(p^{2}n/\log n)$ on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption‐reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work, incorporating the work of Nenadov, together with additional ideas and simplifications.     

PtII-Catalyzed Hydroxylation of Terminal Aliphatic C(sp3)−H Bonds with Molecular Oxygen

The practical application of Shilov-type Pt catalysis to the selective hydroxylation of terminal aliphatic C−H bonds remains a formidable challenge, due to difficulties in replacing Pt^IV with a more economically viable oxidant, particularly O₂. We report the potential of employing FeCl₂ as a suitable redox mediator to overcome the kinetic hurdles related to the direct use of O₂ in the Pt reoxidation. For the selective conversion of butyric acid to γ-hydroxybutyric acid (GHB), a significantly enhanced catalyst activity and stability (turnover numbers (TON)>30) were achieved under 20 bar O₂ in comparison to current state-of-the-art systems (TON<10). In this regard, essential reaction parameters affecting the overall activity were identified, along with specific additives to attain catalyst stability at longer reaction times. Notably, deactivation by reduction to Pt⁰ was prevented by the addition of monodentate pyridine derivatives, such as 2-fluoropyridine, but also by introducing varying partial pressures of N₂ in the gaseous atmosphere. Finally, stability tests revealed the involvement of Pt^II and FeCl₂ in catalyzing the non-selective overoxidation of GHB. Accordingly, in situ esterification with boric acid proved to be a suitable strategy to maintain enhanced selectivities at much higher conversions (TON>60). Altogether, a useful catalytic system for the selective hydroxylation of primary aliphatic C−H bonds with O₂ is presented.