A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.     

Molybdenum Complexes Supported by Mixed NHC/Phosphine Ligands: Activation of N2 and Reaction With P(OMe)3 to the First Meta‐Phosphite Complex

Molybdenum(0) dinitrogen complexes, supported by the mixed NHC/phosphine pincer ligand PCP, exhibit an extreme activation of the N₂ ligand due to a very π‐electron‐rich metal center. The low thermal stability of these compounds can be increased using phosphites instead of phosphines as coligands. Through an amalgam reduction of [MoCl₃(PCP)] in the presence of trimethyl phosphite and N₂ the highly activated and room‐temperature stable dinitrogen complex [Mo(N₂)(PCP)(P(OMe)₃)₂] is obtained. As a second product, the first transition metal complex containing the meta‐phosphite ligand P(O)(OMe) originates from this reaction.     

Time dependent Schrödinger equation for black hole evaporation: No information loss

In 1976 S. Hawking claimed that “Because part of the information about the state of the system is lost down the hole, the final situation is represented by a density matrix rather than a pure quantum state”. ¹ This was the starting point of the popular “black hole (BH) information paradox”.