Abstract |
Let S be a set of k distinct elements. An oriented k coloring of an oriented graph D is a mapping f:V(D)→S such that (i) if xy is conatined in A(D), then f(x)≠f(y) and (ii) if xy,zt are conatined in A(D) and f(x)=f(t), then f(y)≠f(z). The oriented chromatic number Xo(D) of an oriented graph D is defined as the minimum k where there exists an oriented k-coloring of D. For an undirected graph G, let O(G) be the set of all orientations D of G. We define the oriented chromatic number Xo(G) of G to be the maximum of Xo(D) over D conatined by O(G). In this thesis, we determine the oriented chromatic number of complete bipartite graphs and complete k-partite graphs. A grid G(m,n) is a graph with the vertex set V(G(m,n))={(i,j) | 1≦i≦m,1≦j≦n} and the edge set E(G(m,n))={(i,j)(x,y) | (i=x+1 and j=y) or (i=x and j=y+1)}. Fertin, Raspaud and Roychowdhury [3] proved Xo(G(4,5))≧7 by computer programs. Here, we give a proof of Xo(D(5,6)=7 where D(5,6) is the orientation of G(5,6). |