Jun Zhang

Fan (Genghua Fan, Distribution of cycle length in graphs, J. Combin. Theory Ser. B 84 (2002),187--202) proved that if $G$ is a graph with minimum degree $\delta(G)\ge3k$ for any positive integer $k$, then $G$ contains $k+1$ cycles $C_0,C_1,\ldots,C_k$ such that $k+1<|E(C_0)|<|E(C_1)|<\cdots<|E(C_k)|$, $|E(C_i)-E(C_{i-1})|=2$, $1\le i\le k-1$, and $1\le|E(C_k)|-|E(C_{k-1})|\le2$, and furthermore, if $\delta(G)\ge3k+1$, then $|E(C_k)|-|E(C_{k-1})|=2$. In this paper, we generalize Fan's result, show that let $G$ be a graph with minimum degree $\delta(G)\ge3$, for any positive integer $k$ (if $k\ge2$, then $\delta(G)\ge4$), if $d_G(u)+d_G(v)\ge6k-1$ for every pair of adjacent vertices $u,v\in V(G)$, then $G$ contains $k+1$ cycles $C_0,C_1,\ldots,C_k$ such that $k+1<|E(C_0)|<|E(C_1)|<\cdots<|E(C_k)|$, $|E(C_i)-E(C_{i-1})|=2$, $1\le i\le k-1$, and $1\le|E(C_k)|-|E(C_{k-1})|\le2$, and furthermore, if $d_G(u)+d_G(v)\ge6k+1$, then $|E(C_k)|-|E(C_{k-1})|=2$.