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 numberlc (G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper we prove that every planar graph G with girth g and maximum degree Δ has lc (G) = ⌈ frac(Δ, 2) ⌉ + 1 if G satisfies one of the following four conditions: (1) g ≥ 13 and Δ ≥ 3; (2) g ≥ 11 and Δ ≥ 5; (3) g ≥ 9 and Δ ≥ 7; (4) g ≥ 7 and Δ ≥ 13. Moreover, we give better upper bounds of linear chromatic number for planar graphs with girth 5 or 6. © 2008 Elsevier B.V. All rights reserved.
Raspaud, A., & Wang, W. (2009). Linear coloring of planar graphs with large girth. Discrete Mathematics, 309(18), 5678–5686. https://doi.org/10.1016/j.disc.2008.04.032