ON THE ORIENTED CHROMATIC NUMBER OF DENSE GRAPHS

Abstract. Let G be a graph with n vertices, m edges, average degree d, and maximum degree . The oriented chromatic number of G is the maximum, taken over all orientations of G, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which d log n. We prove that every such graph has oriented chromatic num-ber at least p n). In the case that d (2 + ) log n, this lower bound is improved to p m). Through a simple connection with harmonious colourings, we prove a general upper bound of O(pn) on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when G is (c log n)-regular for some constant c> 2, in which case the oriented chromatic number is between p n log n) and O(pn log n)