List edge and list total colorings of planar graphs without short cycles

Abstract

Let G be a planar graph with maximum degree Δ ( G ) . We use χ l ′ ( G ) and χ l ″ ( G ) to denote the list edge chromatic number and list total chromatic number of G, respectively. In this paper, it is proved that χ l ′ ( G ) = Δ ( G ) and χ l ″ ( G ) = Δ ( G ) + 1 if Δ ( G ) ⩾ 6 and G has neither C 4 nor C 6 , or Δ ( G ) ⩾ 7 and G has neither C 5 nor C 6 , where C k is a cycle of length k.