Improvement on Brook theorem for (3 Times K1)-free Graphs

Problem of finding an optimal upper bound for the chromatic no. of a (3 Times K1)-free graph is still open and pretty hard. Here we prove that for a (3 Times K1)-free graph G with maximum degree greater than or equal to 8, {\chi} is less than or equal to max (maximum degree-1, {\omega}). We also prove that if G is (3 Times K1)-free, {\omega} is equal to 4 and maximum degree is greater than or equal to 7, then {\chi} is less than or equal to maximum degree-1. This implies that Borodin & Kostochka Conjecture is true for (3 Times K1)-free graphs as a corollary.