# Colorings and orientations of graphs

## Authors

- Received:
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DOI: 10.1007/BF01204715

- Cite this article as:
- Alon, N. & Tarsi, M. Combinatorica (1992) 12: 125. doi:10.1007/BF01204715

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## Abstract

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)*.