Hamiltonian cycles in cubic Cayley graphs: the 〈2,4k,3〉 case

It was proved by Glover and Maru?i? (J. Eur. Math. Soc. 9:775–787, 2007), that cubic Cayley graphs arising from groups G=〈a,x∣a ²=x ^s =(ax)³=1,…〉 having a (2,s,3)-presentation, that is, from groups generated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4, and have a Hamiltonian path when |G| is congruent to 0 modulo 4. In this article the existence of a Hamiltonian cycle is proved when apart from |G| also s is congruent to 0 modulo 4, thus leaving |G| congruent to 0 modulo 4 with s either odd or congruent to 2 modulo 4 as the only remaining cases to be dealt with in order to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs.     

Nonstoichiometric interpolyelectrolyte complexes as colloidal dispersions based on NaPAMPS and their interaction with colloidal silica particles

Preparation and characterization of some nonstoichiometric interpolyelectrolyte complexes (NIPECs) as stable colloidal dispersions by the interaction between poly(sodium 2-acrylamido-2-methylpropanesulfonate) (NaPAMPS) and three strong polycations bearing quaternary ammonium salt centres in the backbone, poly(diallyldimethylammonium chloride) (PDADMAC) and two polycations containing N,N-dimethyl-2-hydroxypropyleneammonium chloride units (PCA₅ and PCA₅D₁), have been followed in this study as a function of the polycation structure and polyelectrolyte concentration. Complex characteristics were followed by polyelectrolyte titration, turbidity and quasi-ellastic light scattering. Almost monodisperse NIPECs nanoparticles with a good storage stability were prepared when total concentration of polyelectrolyte was varied in the range 0.85-6.35 mmol/L, at a ratio between charges (n⁻/n⁺) of 0.7. NIPECs as a new kind of flocculants were used to flocculate a stable monodisperse silica suspension. The main advantage of NIPECs as flocculants is the broad flocculation window, which is a very important aspect for industrial applications.