A generalization of Fan’s results: Distribution of cycle lengths in graphs

Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187–202] proved that if GG is a graph with minimum degree δ(G)≥3kδ(G)≥3k for any positive integer kk, then GG contains k+1k+1 cycles C0,C1,…,CkC0,C1,…,Ck such that k+1<|E(C0)|<|E(C1)|<⋯<|E(Ck)|k+1<|E(C0)|<|E(C1)|<⋯<|E(Ck)|, |E(Ci)−E(Ci−1)|=2|E(Ci)−E(Ci−1)|=2, 1≤i≤k−11≤i≤k−1, and 1≤|E(Ck)|−|E(Ck−1)|≤21≤|E(Ck)|−|E(Ck−1)|≤2, and furthermore, if δ(G)≥3k+1δ(G)≥3k+1, then |E(Ck)|−|E(Ck−1)|=2|E(Ck)|−|E(Ck−1)|=2. In this paper, we generalize Fan’s result, and show that if we let GG be a graph with minimum degree δ(G)≥3δ(G)≥3, for any positive integer kk (if k≥2k≥2, then δ(G)≥4δ(G)≥4), if dG(u)+dG(v)≥6k−1dG(u)+dG(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G)u,v∈V(G), then GG contains k+1k+1 cycles C0,C1,…,CkC0,C1,…,Ck such that k+1<|E(C0)|<|E(C1)|<⋯<|E(Ck)|k+1<|E(C0)|<|E(C1)|<⋯<|E(Ck)|, |E(Ci)−E(Ci−1)|=2|E(Ci)−E(Ci−1)|=2, 1≤i≤k−11≤i≤k−1, and 1≤|E(Ck)|−|E(Ck−1)|≤21≤|E(Ck)|−|E(Ck−1)|≤2, and furthermore, if dG(u)+dG(v)≥6k+1dG(u)+dG(v)≥6k+1, then |E(Ck)|−|E(Ck−1)|=2|E(Ck)|−|E(Ck−1)|=2.