Local coloring of self complementary graphs

Let G=(V,E) be a graph. 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⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥ms, where ms 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 χℓ(c). The local chromatic number of G is χℓ(G)=min{χℓ(c)}, where the minimum is taken over all local colorings c of G. In this paper we study the local coloring for some self complementary graphs. Also we present a sc-graph with local chromatic number k for any given integer k≥6.