For each vertex \(v\) in a graph \(G\), the maximum length of a cycle which passes through \(v\) is called the cycle number of \(v\), denoted by \(c(v)\). A sequence \(a_1,a_2,\dots,a_n\) of nonnegetive integers is called a cycle sequence of a graph \(G\) if the vertices of \(G\) can be labeled as \(v_1,v_2,\dots,v_n\) such that \(a_i=c(v_i)\) for \(1\leq i\leq n\). The authors give necessary and sufficient conditions for a sequence to be a cycle sequence. A polynomial time procedure for recognizing cycle sequences is derived as well.