Upper bounds on the linear chromatic number of a graph

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic number lc(G) of G is the smallest number of colors in a linear coloring of G.Let G be a graph with maximum degree Δ(G). In this paper we prove the following results: (1) lc(G)≤12(Δ(G)2+Δ(G)); (2) lc(G)≤8 if Δ(G)≤4; (3) lc(G)≤14 if Δ(G)≤5; (4) lc(G)≤⌊0.9Δ(G)⌋+5 if G is planar and Δ(G)≥52.