On linear coloring of planar graphs with small girth ?
A vertex coloring of a graph G 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 ??39 ? ? 39 and girth g?6 g ? 6 is linearly ( ? ? 2 ? + 1 ) -colorable. (2) There exists an integer ?0 ? 0 such that every planar graph with maximum degree ???0 ? ? ? 0 and girth g?5 g ? 5 is linearly ( ? ? 2 ? + 1 ) -colorable. The latter result is best possible in some sense.
linear coloring planar graph girth cycle