Coloring Graphs without Short Non-bounding Cycles

It is shown that there is a constant c such that if G is a graph embedded in a surface of genus g (either orientable or non-orientable) and the length of a shortest non-bounding cycle of G is at least c log(g + 1), then G is six-colorable. A similar result holds for three- and four-colorings under additional assumptions on the girth of G.