A uniform family of tissue P systems with protein on cells solving 3-coloring in linear time

A new variant of tissue P systems called tissue P system with protein on cells is used in this paper. It has the ability to move proteins between cells. It is inspired from the biology that the cells communicate by sending and receiving signals. Signals most often move through the cell by passing from protein to protein. In tissue P systems with protein on cells, multisets of objects together with proteins between cells are exchanged. We present in this paper a linear solution of the 3-coloring problem, a well known NP-complete problem.