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List edge and list total colorings of planar graphs without 4-cycles. (English) Zbl 1108.05038
O. V. Borodin, A. V. Kostochka and D. R. Woodall [J. Comb. Theory, Ser. B 71, 184–204 (1997; Zbl 0876.05032)] proved that if $$G$$ is a simple planar graph with maximum degree $$\Delta \geq 12$$ then the list edge chromatic number $$\chi _{\mathrm{list}}^{\prime }(G)=\Delta$$ and the list total chromatic number $$\chi _{\mathrm{list}}^{\prime \prime }(G)=\Delta +1$$.
In the paper under review these equalities are shown to hold for a planar graph $$G$$ which satisfies one of the following conditions: $$\Delta \geq 7$$ and $$G$$ has no cycle of length 4, $$\Delta =6$$ and $$G$$ has no cycle of length 4 or 5, or $$\Delta =5$$ and $$G$$ has no cycle whose length lies in the closed interval $$[4,8]$$. In addition, $$\chi _{\mathrm{list}}^{\prime }(G)=\Delta$$ is shown to hold for a planar graph $$G$$ with $$\Delta =4$$ if $$G$$ has no cycle whose length lies in the closed interval $$[4,14]$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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##### References:
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