Zhongfu Zhang

Graph theory is a sort of models which can be applied in various science fields such as computer science, physics, biology, chemistry, strategy etc. Graph coloring is one of the chief topics in graph research\cite{5,6,7,8}. In this paper, we introduce definitions of the vertex distinguishing edge coloring, vertex distinguishing equitable edge coloring, $D(\beta)$ vertex distinguishing edge coloring\cite{1}, adjacent vertex distinguishing edge coloring\cite{2}, adjacent vertex distinguishing equitable edge coloring, acyclic edge coloring, vertex distinguishing acyclic edge coloring,adjacent vertex distinguishing acyclic edge coloring, acyclic equitable edge coloring, adjacent vertex distinguishing acyclic equitable edge coloring, incidence edge coloring, equitable total coloring, adjacent vertex distinguishing total coloring\cite{3,4}, adjacent vertex distinguishing equitable total coloring, vertex distinguishing total coloring, vertex distinguishing equitable total coloring, $D(\beta)$ vertex distinguishing total coloring, adjacent vertex distinguishing strong total coloring, vertex distinguishing strong total coloring. And some results, conjecture and unsolved problem are also discussed.