In a cellular mobile communication control system, assignment channel for a call in a cell so as to achieve high spectral efficience is an important problem within limited frequence bandwidth. The spectral efficiency is related to the coloring problem of graph theory in a cellular mobile communication control system. In this paper, we propose channel offset scheme using a graph theory of cellular mobile communication control system and formulate chromatic bandwidth of channel offset system which is related graph coloring problem. From formulated channel assignment problem, we investgate an optimal channel offset scheme of more efficent frequence spectrum and cell design according to channel constitution and give and upper and lower bound for overall srectral bandwidth.