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A structural theorem on planar graphs and its application to coloring. (Russian) Zbl 0760.05034
Let $$G$$ be a planar graph with maximum degree $$\Delta(G)$$. $$\chi_{vef}(G)$$ denotes the minimum number of colours which are needed for colouring the vertices, edges and faces of $$G$$ such that any two incident or adjacent elements receive different colours. The author proves that $$\Delta(G)\geq 8$$ implies $$\chi_{vef}(G)\leq\Delta+4$$. Moreover, a structural theorem on planar graphs is obtained and its applications to colouring problems are discussed.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
planar graph; normal map; colouring