Linear coloring of graphs embeddable in a surface of nonnegative characteristic

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we prove that every graph G with girth g(G) and maximum degree Delta(G) that can be embedded in a surface of nonnegative characteristic has lc(G) = [Delta(G)/2] + 1 if there is a pair (Delta, g) is an element of {(13, 7), (9, 8), (7, 9), (5, 10), (3, 13)} such that G satisfies Delta(G) >= Delta and g(G) >= g.