Linear coloring of planar graphs with large girth

Abstract

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 number lc ( 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 ) = ⌈ 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.