Assume k is a positive integer, $\lambda =\{k_1,k_2,\dots ,k_q\}$ is a partition of k and G is a graph. A λ-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 $C_1\cup C_2\dots \cup C_q$ and for each vertex v of G, $|L(v)\cap C_i|=k_i$. We say G is λ-choosable if for each λ-assignment L of G, G is L-colourable. It follows from the definition that if $\lambda =\{k\}$, then λ-choosable is the same as k-choosable, if $\lambda =\{1,1,\dots ,1\}$, then λ-choosable is equivalent to k-colourable. For the other partitions of k sandwiched between $\{k\}$ and $\{1,1,\dots ,1\}$ in terms of refinements, λ-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions $\lambda ,{\lambda }^{\prime }$ of k, every λ-choosable graph is ${\lambda }^{\prime }$-choosable if and only if ${\lambda }^{\prime }$ is a refinement of λ. Then we study λ-choosability of special families of graphs. The Four Colour Theorem says that every planar graph is $\{1,1,1,1\}$-choosable. A very recent result of Kemnitz and Voigt implies that for any partition λ of 4 other than $\{1,1,1,1\}$, there is a planar graph which is not λ-choosable. We observe that, in contrast to the fact that there are non-4-choosable 3-chromatic planar graphs, every 3-chromatic planar graph is $\{1,3\}$-choosable, and that if G is a planar graph whose dual $G^{⁎}$ has a connected spanning Eulerian subgraph, then G is $\{2,2\}$-choosable. We prove that if n is a positive even integer, λ is a partition of $n-1$ in which each part is at most 3, then $K_n$ is edge λ-choosable. Finally we study relations between λ-choosability of graphs and colouring of signed graphs and generalized signed graphs. A conjecture of Máčajová, Raspaud and Škoviera that every planar graph is signed 4-colcourable is recently disproved by Kardoš and Narboni. We prove that every signed 4-colourable graph is weakly 4-choosable, and every signed $Z_4$-colourable graph is $\{1,1,2\}$-choosable. The later result combined with the above result of Kemnitz and Voigt disproves a conjecture of Kang and Steffen that every planar graph is signed $Z_4$-colourable. We shall show that a graph constructed by Wegner in 1973 is also a counterexample to Kang and Steffen's conjecture, and present a new construction of a non-$\{1,3\}$-choosable planar graphs.

假设k是一个正整数，$\lambda =\{{ķ}_{\mathrm{1个}}，{ķ}_2，\dots ，{ķ}_q\}$是一个分区ķ和G ^是一个曲线图。甲λ的-assignment ģ是ķ -assignment大号的ģ使得颜色集合${\bigcup }_{v\in V（G）}\mathrm{大号}（v）$可以分为q个子集$C_{\mathrm{1个}}\cup C_2\dots \cup C_q$并为每个顶点v的ģ，$|\mathrm{大号}（v）\cap C_{\mathrm{一世}}|={ķ}_{\mathrm{一世}}$。我们说摹是λ -choosable如果每个λ -assignment大号的摹，摹是大号-colourable。从定义可以得出，如果$\lambda =\{ķ\}$，则λ- choosable与k -choosable相同，如果$\lambda =\{\mathrm{1个}，\mathrm{1个}，\dots ，\mathrm{1个}\}$，则λ- chooseable等效于k -colorable。对于k之间的其他分区$\{ķ\}$ 和 $\{\mathrm{1个}，\mathrm{1个}，\dots ，\mathrm{1个}\}$在细化方面，λ-选择能力揭示了图的可着色性的复杂层次。我们证明对于两个分区$\lambda ，{\lambda }^{\prime }$的k，每个λ-选择图是${\lambda }^{\prime }$-仅当且仅当 ${\lambda }^{\prime }$是λ的细化。然后，我们研究图的特殊族的λ选择性。四色定理说每个平面图都是$\{\mathrm{1个}，\mathrm{1个}，\mathrm{1个}，\mathrm{1个}\}$-可选择。Kemnitz和Voigt的最新结果表明，对于除4以外的任何λ分区$\{\mathrm{1个}，\mathrm{1个}，\mathrm{1个}，\mathrm{1个}\}$，有一个平面图，它不是λ选择的。我们观察到，与存在非4选择的3色平面图相反，每个3色平面图都是$\{\mathrm{1个}，3\}$-可选择的，如果G是一个平面图，其对偶$G^{⁎}$有一个连通的跨欧拉子图，那么G是$\{2，2\}$-可选择。我们证明如果n是一个正偶数整数，则λ是$ñ-\mathrm{1个}$ 其中每个部分最多为3，则 ${ķ}_{ñ}$可以选择边λ。最后，我们研究了图的λ-选择性与有符号图和广义有符号图的着色之间的关系。Máčajová，Raspaud和Škoviera的一个猜想是，每个平面图都是4色签名的，最近Kardoš和Narboni对此进行了反驳。我们证明每个带符号的4色图都是弱4可选择的，并且每个带符号的${ž}_4$-彩色图形是 $\{\mathrm{1个}，\mathrm{1个}，2\}$-可选择。后面的结果与Kemnitz和Voigt的上述结果相结合，反证了Kang和Steffen的猜想，即每个平面图都是有符号的${ž}_4$-彩色的。我们将证明，韦格纳（Wegner）在1973年构造的图也是康和斯蒂芬猜想的反例，并提出了一种非$\{\mathrm{1个}，3\}$-可选择的平面图。