For a collection of finite graphs, the bounded chromatic number χ_B() is the smallest number of colors c for which there exists an integer N such that every graph G can be vertex c-colored without forcing more than N monochromatic edges. The bounded (simple) path chromatic number _χP() (_χSP()) is the smallest number of colors c for which there exists an integer N such that every graph G can be c-colored without forcing a monochromatic (simple) path of length more than N. For the set _g of all graphs of genus g it is known that 4≤_χB(_g)≤6, and _χSP(_g)=4. In this paper we show that _χB(_g)≤5, and _χP(_g)=4. For g≥1, let μ₅(g) (π₄(g)) denote the smallest integer x such that every graph G_g can be 5-colored without forcing more than x monochromatic edges (4-colored without forcing a monochromatic path of length more than x). We also show that 2g ≤ μ₅(g) ≤ 74g − 36 and .