Linear coloring of planar graphs with large girth

A proper vertex coloring of a graph GG 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 GG is the smallest number of colors in a linear coloring of GG. In this paper we prove that every planar graph GG with girth gg and maximum degree ΔΔ has lc(G)=⌈Δ2⌉+1 if GG satisfies one of the following four conditions: (1) g≥13g≥13 and Δ≥3Δ≥3; (2) g≥11g≥11 and Δ≥5Δ≥5; (3) g≥9g≥9 and Δ≥7Δ≥7; (4) g≥7g≥7 and Δ≥13Δ≥13. Moreover, we give better upper bounds of linear chromatic number for planar graphs with girth 5 or 6.