For several problems, restricting attention to special classes of graphs has yielded better algorithms. In particular, restricting to planar graphs yields efficient parallel algorithms for several graph problems. In this paper we extend these algorithms to $K_{3}$,${3}$-free graphs, showing that the restriction of planarity is not important. The three problems dealt with are: graph coloring, depth first search and maximal independent sets. As a corollary we show that $K{3}$,$_{3}$-free graphs are five colorable (this bound is tight).