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A refinement of choosability of graphs. (English) Zbl 1430.05046
Summary: Assume $$k$$ is a positive integer, $$\lambda = \{k_1, k_2, \ldots, k_q\}$$ 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 $$C_1 \cup C_2 \ldots \cup C_q$$ 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. It follows from the definition that if $$\lambda = \{k\}$$, then $$\lambda$$-choosable is the same as $$k$$-choosable, if $$\lambda = \{1, 1, \ldots, 1\}$$, then $$\lambda$$-choosable is equivalent to $$k$$-colourable. For the other partitions of $$k$$ sandwiched between $$\{k\}$$ and $$\{1, 1, \ldots, 1 \}$$ in terms of refinements, $$\lambda$$-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions $$\lambda, \lambda^\prime$$ of $$k$$, every $$\lambda$$-choosable graph is $$\lambda^\prime$$-choosable if and only if $$\lambda^\prime$$ is a refinement of $$\lambda$$. Then we study $$\lambda$$-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 A. Kemnitz and M. Voigt [Electron. J. Comb. 25, No. 2, Research Paper P2.46, 5 p. (2018; Zbl 1388.05048)] implies that for any partition $$\lambda$$ of 4 other than $$\{1, 1, 1, 1 \}$$, there is a planar graph which is not $$\lambda$$-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 $$G^\ast$$ has a connected spanning Eulerian subgraph, then $$G$$ is $$\{2, 2 \}$$-choosable. We prove that if $$n$$ is a positive even integer, $$\lambda$$ is a partition of $$n - 1$$ in which each part is at most 3, then $$K_n$$ is edge $$\lambda$$-choosable. Finally we study relations between $$\lambda$$-choosability of graphs and colouring of signed graphs and generalized signed graphs. A conjecture of E. Máčajová et al. [Electron. J. Comb. 23, No. 1, Research Paper P1.14, 10 p. (2016; Zbl 1329.05116)] that every planar graph is signed 4-colcourable is recently disproved by F. Kardoš and J. Narboni [“On the 4-color theorem for signed graphs ”, Preprint, arXiv:1906.05638]. 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 Y. Kang and E. Steffen [J. Graph Theory 87, No. 2, 135–148 (2018; Zbl 1383.05103)] that every planar graph is signed $$Z_4$$-colourable. We shall show that a graph constructed by G. Wegner [Isr. J. Math. 14, 409–412 (1973; Zbl 0265.05104)] is also a counterexample to Kang and Steffen’s conjecture, and present a new construction of a non-$$\{1, 3 \}$$-choosable planar graphs.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C22 Signed and weighted graphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05A17 Combinatorial aspects of partitions of integers
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