 
Summary: COLORINGS AND ORIENTATIONS OF GRAPHS
N. Alon*
and
M. Tarsi
Departments of Pure Mathematics
and Computer Science
School of Mathematical Sciences
Raymond and Beverly Sackler
Faculty of Exact Sciences
Tel Aviv University
Tel Aviv, Israel
* Research supported in part by a United StatesIsrael BSF Grant and by a Bergmann Memorial
Grant. AMS Subject Classification codes: 05C15, 05C20
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 vertexcoloring of G assigning to each vertex v a color from S(v).
