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 G is a graph with minimum degree δ ( G ) ≥ 3 k for any positive integer k , then G contains k + 1 cycles C 0 , C 1 , … , C k such that k + 1 < | E ( C 0 ) | < | E ( C 1 ) | < ⋯ < | E ( C k ) | , | E ( C i ) − E ( C i − 1 ) | = 2 , 1 ≤ i ≤ k − 1 , and 1 ≤ | E ( C k ) | − | E ( C k − 1 ) | ≤ 2 , and furthermore, if δ ( G ) ≥ 3 k + 1 , then | E ( C k ) | − | E ( C k − 1 ) | = 2 . In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ ( G ) ≥ 3 , for any positive integer k (if k ≥ 2 , then δ ( G ) ≥ 4 ), if d G ( u ) + d G ( v ) ≥ 6 k − 1 for every pair of adjacent vertices u , v ∈ V ( G ) , then G contains k + 1 cycles C 0 , C 1 , … , C k such that k + 1 < | E ( C 0 ) | < | E ( C 1 ) | < ⋯ < | E ( C k ) | , | E ( C i ) − E ( C i − 1 ) | = 2 , 1 ≤ i ≤ k − 1 , and 1 ≤ | E ( C k ) | − | E ( C k − 1 ) | ≤ 2 , and furthermore, if d G ( u ) + d G ( v ) ≥ 6 k + 1 , then | E ( C k ) | − | E ( C k − 1 ) | = 2 .