A 4-COLOR THEOREM FOR SURFACES OF GENUS g

For .f a set of graphs, define the bounded chromatic number Xb(&) (resp. the bounded path chromatic number Xpfë) ) to be the minimum number of colors c for which there exists a constant M such that every graph G € 2? can be vertex c-colored so that all but M vertices of G are properly colored (resp. the length of the longest monochromatic path in G is at most M ). For 3? the set of toroidal graphs, Albertson and Stromquist [1] conjectured that the bounded chromatic number is 4. For any fixed g > 0 , let S'g denote the set of graphs of genus g . The Albertson-Stromquist conjecture can be extended to the conjecture that XB^g) = 4 for all g > 0. In this paper we show that 4 < Xßi^g) < 6. We also show that the bounded path chromatic number Xpi-^s) equals 4 for all g>0. Let fic(g,n)(nc{g,n)) denote the minimum / such that every graph of genus g on n vertices can be c-colored without forcing / + 1 monochromatic edges (a monochromatic path of length / + 1 ). We also obtain bounds for fic(g,n) and nc(g. n).