Linear coloring of sparse graphs ?
A linear k k -coloring  of a graph G G is a proper k k -coloring of G G such that the graph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic number  lc ( G ) of a graph G G is the smallest number k k such that G G has a linear k k -coloring. In this paper, we prove that if G G is a graph with maximum degree ? ? and maximum average degree m a d ( G ) < 14 5 , then lc ( G ) ? ? ? 2 ? + 2 . In particular, lc ( G ) ? ? ? 2 ? + 2 if G G is a planar graph with maximum degree ? ? and girth g(G)?7 g ( G ) ? 7 . This improves some known results in this direction, and further supports a conjecture recently proposed by D. Cranston and G. Yu, which states that lc ( G ) ? ? ? 2 ? + 2 for every graph G G with mad(G)<3 m a d ( G ) < 3 .
linear coloring maximum degree maximum average degree girth