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Colorings and orientations of graphs. (English) Zbl 0756.05049
Summary: Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: If $$G$$ is a directed graph with maximum outdegree $$d$$, and if the number of Eulerian subgraphs of $$G$$ with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a set $$S(v)$$ of $$d+1$$ colors for each vertex $$v$$ of $$G$$ there is a legal vertex-coloring of $$G$$ assigning to each vertex $$v$$ a color from $$S(v)$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C20 Directed graphs (digraphs), tournaments
##### Keywords:
orientations; chromatic number; vertex-coloring
Full Text:
##### References:
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