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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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