On partial order of λ-choosability for planar graphs

For a positive integer k, let λ = k 1, k 2, …, k j be a multiset of positive integers with ∑ i = 1 j k i = k. A k -assignment L of a graph G is a λ-assignment if the color set ⋃ v ∈ V (G) L (v) can be partitioned into color classes C 1, C 2, …, C j such that | L (v) ∩ C i | = k i for each vertex v and each 1 ≤ i ≤ j. We say that G is λ-choosable if it admits a coloring using colors of L for every λ -assignment L. An operation called a refinement of λ naturally defines a partial order ⩽ on the multisets of positive integers, and ⩽ transmits to λ -choosability of graphs, i. e. , for two multisets of positive integers λ and λ ′, λ ′ ⩽ λ implies that every λ -choosable graph is λ ′ -choosable. In this paper, we focus on the differences between 1, 3 -choosability and 2, 2 -choosability of planar graphs, where 1, 3 and 2, 2 are incomparable under ⩽. In particular, we observe that the non- 1, 3 -choosable planar graphs constructed by Zhu (2020) are 2, 2 -choosable. As a counterpart, we construct an infinite family of planar graphs which are 1, 3 -choosable but not 2, 2 -choosable. Consequently, we clarify the distinction among λ -choosable planar graphs.