List strong linear 2-arboricity of sparse graphs

Abstract

A graph G is called (k, j)-colorable if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[Vi] induced by the vertices of Vi has maximum degree at most k for i = 1 and at most j for i = 2. In particular, Havet and Sereni [2006] proved that each planar graph G is list (1, 1)-colorable if its girth, g(G), is at least 8 and list (2, 2)-colorable if g(G)⩾6. Borodin et al. [2009] proved that every planar graph is (2, 1)-colorable if g(G)⩾7 and (5, 1)-colorable if g(G)⩾6. In fact, all these results were proved for each not necessarily planar sparse graph G, i.e. having a low maximum average degree, mad(G), over all subgraphs. A graph is a strong linear forest if its every connected component is a path of at most three vertices. Note that at most one third of the vertices in a strong linear forest have degree 2. We prove that each planar graph G with g(G)⩾7 can be partitioned into two strong linear forests. The same is true for each graph G with g(G)⩾7 and **image**, and we actually prove a choosability version of this result. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:83-90, 2011 © 2011 Wiley Periodicals, Inc.