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 . be a graph with maximum degree Δ ( G ) . In this paper we prove the following results: (1) lc ( G ) ≤ 1 2 ( Δ ( G ) 2 + Δ ( G ) ) ; (2) lc ( G ) ≤ 8 if Δ ( G ) ≤ 4 ; (3) lc ( G ) ≤ 14 if Δ ( G ) ≤ 5 ; (4) lc ( G ) ≤ ⌊ 0.9 Δ ( G ) ⌋ + 5 if G is planar and Δ ( G ) ≥ 52 .