Colorings and orientations of graphs

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).