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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
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