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The chromatic number of a graph of girth 5 on a fixed surface. (English) Zbl 1020.05030
In this interesting paper it is proved that, for every surface $$S$$ and every natural number $$k$$, there exists a natural number $$f(S,k)$$ such that the following holds: If $$G$$ is a graph of girth 5 on $$S$$, and $$H$$ is a 3-colored subgraph with at most $$k$$ vertices, then either the coloring of $$H$$ can be extended to a 3-coloring of $$G$$, or else there is a small obstruction containing $$H$$, that is, a subgraph $$H'$$ with at most $$f(S,k)$$ vertices such that the coloring of $$H$$ cannot be extended to a 3-coloring of $$H'$$. In particular, there are only finitely many 4-color-critical graphs of girth 5 on $$S$$, as a 4-color-critical graph of girth 5 on $$S$$ has at most $$f(S,1)$$ vertices. It follows that, if $$G$$ is a graph of girth 5 on $$S$$, and all noncontractible cycles in $$G$$ have length greater than $$f(S,1)$$, then $$G$$ is 3-colorable. The result is best possible in the sense that there are infinitely many 4-color-critical graphs of girth 4 on $$S$$, except when $$S$$ is the sphere. As a consequence, it is deduced that the chromatic number of graphs of girth 5 on $$S$$ can be found in polynomial time.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
4-color-critical graph; polynomial time
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##### References:
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