On the oriented chromatic number of graphs with given excess

The excess of a graph G is defined as the minimum number of edges that must be deleted from G in order to get a forest. We prove that every graph with excess at most k   has chromatic number at most 12(3+1+8k) and that this bound is tight. Moreover, we prove that the oriented chromatic number of any graph with excess k   is at most k+3k+3, except for graphs having excess 1 and containing a directed cycle on 5 vertices which have oriented chromatic number 5. This bound is tight for k⩽4k⩽4.