Girth and Λ $\lambda $‐choosability of Graphs

Assume k is a positive integer, A = {k1, k2, ..., kq} is a partition of k and G is a graph. A A-assignment of G is a k-assignment L of G such that the colour set ?vE( V ) GL (v) can be partitioned into q subsets C1 U C2 U center dot center dot center dot U Cq and for each vertex v of G, ?L (v) n Ci? = ki. We say G is A-choosable if for each A-assignment L of G, G is L-colourable. In particular, if A = {k}, then A-choosable is the same as k-choosable, and if A = {1, 1, ..., 1}, then A-choosable is equivalent to k-colourable. For the other partitions of k sandwiched between {k} and {1, 1, ..., 1} in terms of refinements, A-choosability reveals a complex hierarchy of colourability of graphs. Assume A = {k1, ..., kq} is a partition of k and A ' is a partition of k ' > k. We write A < A ' if there is a partition A '' = {k1 '', ..., kq ''} of k ' with ki '' > ki for i = 1, 2, ..., q and A ' is a refinement of A ''. It follows from the definition that if A < A ', then every A-choosable graph is A '-choosable. It was proved in Zhu that the converse is also true. This paper strengthens this result and proves that for any A A ', for any integer g, there exists a graph of girth at least g which is A-choosable but not A '-choosable.