On the equality of the grundy and ochromatic numbers of a graph

It is proved in this note that the Grundy number, T(G), and the ochromatic number, x’(G), are the same for any graph G. An n-coloring of a graph G = (V, E) is a function f from I/ onto N ={1,2,..., n} such that, whenever vertices u and u are adjacent. then f(u) f f(u). An n-coloring is complete if for every pair i,j of integers, 1 5 i 5 j 5 n, there exist a pair U, u of adjacent vertices such that f(u) = i and f(v) = j. The chromatic number, x(G), and the achromutic number, I/J(G), are the smallest and largest values n, respectively, for which G has a complete n-coloring. A complete n-coloring g : V ---, N is a Grundy n-coloring if. for every vertex u E V, g(v) is the smallest integer that is not assigned to any vertex adjacent to II. The Grundy number, T(G), is the largest n for which G has a Grundy n-coloring. Finally we define a parsimonious proper coloring (ppc). Let 4: u,, v?, . , u,, be an arbitrary ordering of the vertices V of graph G = (V, E). Consider coloring the vertices of G in the following manner: the vertices are colored in the given order $; when a vertex Ye is to be colored. it must be assigned one of the colors that has been used to color the vertices u, , . , v,-, provided a valid coloring will result; only if u, is adjacent to a vertex of every currently used color can a new color be assigned; if V, can be assigned more than one color, one must choose a color that results in the least number of colors being used to color G. The minimum number of colors used to color G in this way, for the Journal of Graph Theory, Vol. 11, No 2. 157-159 (1987)