Good and Semi-Strong Colorings of Oriented Planar Graphs

A k?coloring of an oriented graph G = (V; A) is an assignment c of one of the colors 1; 2; : : :; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6 = c(y). The k?coloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z). A k?coloring is said to be semi?strong if for every vertex x of G, c(z) 6 = c(t) for any pair fz; tg of vertices of N ? (x). We show that every oriented planar graph has a good coloring using at most 5 2 4 colors and that every oriented planar graph G = (V; A) with d ? (x) 3 for every x 2 V has a good and semi-strong coloring using at most 4 5 2 4 colors.