关键词: 原子圈, 分解, 合并, 单条公共边连通, 充分必要条件, 必要条件计算公式

Abstract: As an NP-complete problem，Hamiltonian graph problem is one of the main unresolved problems in graph theory.In 1968 Grinberg advanced a necessary condition for Hamiltonian planar graphs，which enhanced the solution of non-Hamiltonian planar graphs and led to a series of research work on 3-regular 3-connected non-Hamiltonian planar graphs.Based on the characteristic of undirected Hamiltonian graph，some new notions about decomposition，mergence and connection in common edge of basic cycles，as well as atomic cycle are defined.Any a connected simple undirected graph G is a Hamiltonian graph if and only if either the Hamiltonian cycle itself is an atomic cycle which contains every vertex of G or the Hamiltonian cycle can always be decomposed into several atomic cycles which are connected with one another by one common edge in a certain order.A new necessary condition formula is derived from this sufficient and necessary condition which can deal with not only planar graphs but also non-planar graphs，even more those planar graphs which can not be treated by Grinberg condition.Moreover，experimental results on some real cases demonstrate the effectiveness of this condition and its formula.

Key words: atomic cycle, decomposition, mergence, connection in one common edge, sufficient and necessary condition, formula of necessary condition