设$ G $是一个$ n $阶图, 若对于每一个$ k\;(3\leqslant k\leqslant n) $, 图$ G $都含有$ k $-圈, 则称图$ G $为泛圈图. 泛圈图是圈理论研究中的重要课题. 研究得到了Hamilton圈上两个不相邻的点在圈上的距离是3的泛圈性结果.
An $ n $-vertex graph is called pancyclic if it contains a cycle of length $ k $ for every $ k\;(3\leqslant k\leqslant n) $. Pancyclic graphs are an important topic in cycle theory. In this paper, we demonstrate pancyclicity by showing that the distance between two non-adjacent vertices on a Hamiltonian cycle is 3.
[1] BONDY J A, MURTY U S R. Graph Theory with Applications [M]. New York: Macmilan Ltd Press, 1976.
[2] BONDY J A. Pancyclic graphs I [J]. Combin Theory (B), 1971, 11(1): 80-84. DOI: 10.1016/0095-8956(71)90016-5.
[3] ZHAO K W, LIN Y, ZHANG P. A sufficient condition for pancyclic graphs [J]. Information Processing Letters, 2009, 109(17): 991-996. DOI: 10.1016/j.ipl.2009.05.009.
[4] ZAMFIRESCU C T. (2)-Pancyclic graphs [J]. Discrete Mathematics, 2013, 161(7/8): 1128-1136.
[5] 刘少强, 陈锦丽. (3)-泛圈图的一些必要条件 [J]. 闽南师范大学学报, 2014, 27(1): 7-15
[6] 陈耀静. (4)-泛圈图的一个必要条件 [J]. 闽南师范大学学报, 2019, 32(1): 10-20
[7] SCHMEICHEL E F, HAKIMI S L. A cycle structure theorem for Hamiltonian graphs [J]. Combin Theory (B), 1988, 45(1): 99-107. DOI: 10.1016/0095-8956(88)90058-5.
[8] REN H. Another cycle structure theorem for Hamiltonian graphs [J]. Discrete Mathematics, 1999, 199(1/2/3): 237-243.