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.