Abstract: For a graph G, c is a proper vertex coloring of G. If every color appears at most k-1 times in the neighbors of each vertex v, then c is called a k-frugal coloring of G. There are two conclusions on the k-frugal list coloring of planar graphs without 4 and 5-cycles:(1)For each planar graph without 4 and 5-cycles, if its maximum degree is Δ and Δ≥3k+8, then the k-frugal list chromatic number is less than or equal to 「(Δ)/(k-1)+2; (2)For each planar graph without 4 and 5-cycles, its k-frugal list chromatic number is no more than 「(Δ)/(k-1)+5.

Key words: frugal list coloring, planar graph, cycle