Girth and $\lambda$-choosability of graphs

Assume $k$ is a positive integer, $\lambda=\{k_1,k_2,...,k_q\}$ is a partition of $k$ and $G$ is a graph. A $\lambda$-assignment of $G$ is a $k$-assignment $L$ of $G$ such that the colour set $\bigcup_{v\in V(G)} L(v)$ can be partitioned into $q$ subsets $C_1\cup C_2\cup\cdots\cup C_q$ and for each vertex $v$ of $G$, $|L(v)\cap C_i|=k_i$. We say $G$ is $\lambda$-choosable if for each $\lambda$-assignment $L$ of $G$, $G$ is $L$-colourable. In particular, if $\lambda=\{k\}$, then $\lambda$-choosable is the same as $k$-choosable, if $\lambda=\{1, 1,...,1\}$, then $\lambda$-choosable is equivalent to $k$-colourable. For the other partitions of $k$ sandwiched between $\{k\}$ and $\{1, 1,...,1\}$ in terms of refinements, $\lambda$-choosability reveals a complex hierarchy of colourability of graphs. Assume $\lambda=\{k_1, \ldots, k_q\}$ is a partition of $k$ and $\lambda'$ is a partition of $k'\ge k$. We write $\lambda\le \lambda'$ if there is a partition $\lambda''=\{k''_1, \ldots, k''_q\}$ of $k'$ with $k''_i \ge k_i$ for $i=1,2,\ldots, q$ and $\lambda'$ is a refinement of $\lambda''$. It follows from the definition that if $\lambda\le \lambda'$, then every $\lambda$-choosable graph is $\lambda'$-choosable. It was proved in [X. Zhu, A refinement of choosability of graphs, J. Combin. Theory, Ser. B 141 (2020) 143 - 164] that the converse is also true. This paper strengthens this result and proves that for any $\lambda\not\le \lambda'$, for any integer $g$, there exists a graph of girth at least $g$ which is $\lambda$-choosable but not $\lambda'$-choosable.