New upper bounds on linear coloring of planar graphs

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, it is proved that every planar graph G with girth g and maximum degree . has (1) lc(G) . . + 21 if . . 9; (2) lc(G)...2.+7 if g . 5; (3) lc(G)...2.+2 if g . 7 and .. 7.