Strange Behavior on DS18B20 Temp sensor and Pull-up Resistor

I mentioned the star-configuration because that one can easily dedicate a digital pin per DS18B20.

You could draw your Arduino board with all the sensors and wiring and distances on a piece of paper and upload a photo of it.

According to the datasheet of the DS18B20, a resistor is 4k7 is a good value to start with and the maximum sink current is at least 4mA. That means that a resistor of 1250 Ω is the lowest allowed value.
They don't give a number for the "typical" current that the DS18B20 can sink. That is probably higher.