Local colorings of graphs

A local coloring of a graph G of order at least 2 is a function c : V(G) -> N having the property that for each set S subset of V (G) with 2 <= vertical bar S vertical bar <= 3, there exist vertices u, v is an element of S such that vertical bar c(u) - c(v)vertical bar >= m(S), where m(S) is the size of the induced subgraph < S >. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by chi e(c). The local chromatic number of G is chi e(G) = min{chi e(c)}, where the minimum is taken over all local colorings c of G. The local chromatic numbers of all complete multipartite graphs are determined.