A result on linear coloring of planar graphs

A proper vertex coloring of a graph is called linear if the subgraph induced by the vertices colored by any two colors is a set of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by lc(G), is the minimum number of colors in a linear coloring of G. In this paper, we show lc(G)⩽Δ(G)+7 for a planar graph G with maximum degree Δ(G).