Linear choosability of sparse graphs

A linear coloring is a proper coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list chromatic number, denoted lc ℓ ( G ) , of sparse graphs. The maximum average degree of a graph G , denoted m a d ( G ) , is the maximum of the average degrees of all subgraphs of G . It is clear that any graph G with maximum degree Δ ( G ) satisfies lc ℓ ( G ) ≥ ⌈ Δ ( G ) / 2 ⌉ + 1 . In this paper, we prove the following results: (1) if mad ( G ) < 12 / 5 and Δ ( G ) ≥ 3 , then lc ℓ ( G ) = ⌈ Δ ( G ) / 2 ⌉ + 1 , and we give an infinite family of examples to show that this result is best possible; (2) if mad ( G ) < 3 and Δ ( G ) ≥ 9 , then lc ℓ ( G ) ≤ ⌈ Δ ( G ) / 2 ⌉ + 2 , and we give an infinite family of examples to show that the bound on mad ( G ) cannot be increased in general; (3) if G is planar and has girth at least 5, then lc ℓ ( G ) ≤ ⌈ Δ ( G ) / 2 ⌉ + 4 .