On some new graph colouring parameters

Some new graph colouring parameters are investigated. The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colouring. It is known that the zero forcing number is an upper bound for the path cover number of graphs. We show that these two parameters coincide for some families of graphs. Correspondingly, we prove that the positive zero forcing number is an upper bound for the tree cover number of graphs. Using this fact we show via a purely graph theoretical approach that the positive zero forcing number of any outerplanar graph equals the tree cover number of that graph. We finally establish a connection between the positive zero forcing number and the chromatic number of a graph.