On linear coloring of planar graphs with small girth

Abstract A vertex coloring of a graph G is linear if the subgraph induced by the vertices of any two color classes is the union of vertex-disjoint paths. In this paper, we study the linear coloring of graphs with small girth and prove that: (1) Every planar graph with maximum degree Δ ≥ 39 and girth g ≥ 6 is linearly ( ⌈ Δ 2 ⌉ + 1 ) -colorable. (2) There exists an integer Δ 0 such that every planar graph with maximum degree Δ ≥ Δ 0 and girth g ≥ 5 is linearly ( ⌈ Δ 2 ⌉ + 1 ) -colorable. The latter result is best possible in some sense.