Electronic – the optimal battery charging speed

Technically the minimum amount of voltage for charging will be anything above the current state of charge. But that's probably not the answer you're looking for, from Lithium-ion battery on Wikipedia:

Lithium-ion is charged at approximately 4.2 ± 0.05 V/cell except for "military long life" that uses 3.92 V to extend battery life. Most protection circuits cut off if voltage greater than 4.3 V or temperature greater than 90 °C is reached. Below 2.50 V/cell the battery protection circuit may render the battery unchargeable with regular charging equipment. Most battery circuits stop at 2.7–3.0 V/cell.

So to achieve a full state of charge you'd normally want to aim at 4.2V. In practice charging Li-Ion safely and efficiently does involve quite a few steps so you may want to look at a dedicated charger chip. You'd need to check the datasheets for any that look of interest but many will operate properly with a supply voltage only a little above 4.2V.

The 3.7V above sounds like the nominal voltage which is the area where the battery will spend most of it's time during the charge to discharge cycle. But they will start out at around 4.2V and drop to a voltage below that. Letting them drop in voltage too far will cause problems, but you'll get some useful life below 3.7V as well.

Look closely at the top plot and you can see it is just applying a simple maximum current OR voltage scheme. Up to about .9 hour the maximum current (looks like about 700 mA) causes lower than the maximum voltage, so the battery is charged with that current. When the battery reaches the 4.2 V limit, the charger holds the voltage there and the current then decreases. Eventually the battery gets so full that the charging current tapers off to nearly 0.

The charger puts out either 700 mA or 4.2 V, whichever results in lower voltage (or current). The battery then decides the other of the two.