A power coloring of a graph G is a mapping c : V (G) → N that assigns colors to the vertices of G in such a way that adjacent vertices are labeled with different colors. A coloring c is called a 4-local coloring if for every subset S ⊆ V (G), with 2 ≤ |S| ≤ 4 there are two vertices u, v such that the difference between colors of u and v, is greater than or equal to the number of edges in the subgraph induced by S. That is, ∀S∃u,v∈ S∋&vbm0;cu-c v&vbm0;≥ms, where ms is the number of edges subgraph induced by S, ms = |E()|. The maximum color assigned by a local coloring c to a vertex of G is c4c= maxcv &vbm0;v∈VG . The four local chromatic numbor of G is defined as c4 G=minc4 c=min &cubl0;max&cubl0;cG&vbm0; wherec isa4-local coloringofG&cubr0;&cubr0;. In this thesis, the four local chromatic number of some well known class of graphs will be determined
