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From the plane to higher surfaces. (English) Zbl 1244.05075
Summary: We show that Grötzsch’s theorem extends to all higher surfaces in the sense that every triangle-free graph on a surface of Euler genus \(g\) becomes 3-colorable after deleting a set of at most \(1000 \cdot g\cdot f(g)\) vertices where \(f(g)\) is the smallest edge-width which guarantees a graph of Euler genus \(g\) and girth 5 to be 3-colorable.
We derive this result from a general cutting technique which we also use to extend other results on planar graphs to higher surfaces in the same spirit, even after deleting only \(1000g\) vertices. These include the 5-list-color theorem, results on arboricity, and various types of colorings, and decomposition theorems of planar graphs into two graphs with prescribed degeneracy properties.
It is not known if the 4-color theorem extends in this way.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
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