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For 9 a set of graphs, define the bounded chromatic number XB(C) (resp. the bounded path chromatic number Xp() ) to be the minimum number of colors c for which there exists a constant M such that every graph G E 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 9 the set of toroidal graphs, Albertson and Stromquist [1] conjectured that the bounded chromatic number is 4. For any fixed g > 0, let 5g denote the set of graphs of genus g . The Albertson-Stromquist conjecture can be extended to the conjecture that XB(5?g) = 4 for-all g > 0. In this paper we show that 4 0 . Let uc (g, n)(7rc (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 ,uc(g, n) and lrc(g, n) .

A 4-color theorem for surfaces of genus 𝑔