The properties of phone cameras vary a lot. With some you can set certain values, with others you can't and just shoot in an "auto" mode. Just the pixel count and the operating system aren't enough to tell this.
There are some Android phones like Oppos N1 where you can set longer exposures up to 8 seconds. With the light painting as the main subject the quality of the resulting images may be high enough. But most phone cameras can't do such long exposures.
If your built in camera app doesn't offer you to set the exposure time you may have a look around into apps that may offer this (e.g. for me CameraFX offers all the few settings that my phone camera hardware offers).
If your phone camera and software don't allow you to do long exposures you can fake the result. You can make a video of the light painting, extract the frames, and combine them to a resulting image with e.g. Luminance HDR or Photoshop/GIMP. You can even have a look into interval camera apps and see if they in combination with your hardware allow a fast enough burst; this would probably yield higher quality frames than video extraction. The mentioned CameraFX has a burst mode that promises a photo every 0.2 seconds but that fails completely with my hardware.