tailieunhanh - Báo cáo toán học: " A [k, k + 1]-Factor Containing A Given Hamiltonian Cycle"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A [k, k + 1]-Factor Containing A Given Hamiltonian Cycle. | A k k 1 -Factor Containing A Given Hamiltonian Cycle Cai Mao-cheng Yanjun Li Institute of Systems Science Chinese Academy of Sciences Beijing 100080 . China caimc@ cogt@ Mikio Kano Department of Computer and Information Sciences Ibaraki University Hitachi 316 Japan kano@ Abstract We prove the following best possible result. Let k 2 be an integer and G be a graph of order n with minimum degree at least k. Assume n 8k 16 for even n and n 6k 13 for odd n. If the degree sum of each pair of nonadjacent vertices of G is at least n then for any given Hamiltonian cycle C of G G has a k k 1 -factor containing C. Submitted December 15 1997 Accepted November 27 1998. MR Subject Number 05C75 Keywords k k 1 -factor Hamiltonian cycle degree condition 1 Introduction All graphs under consideration are undirected finite and simple. A graph G consists of a non-empty set V G of vertices and a set E G of edges. For two vertices x and y of G let xy and yx denote an edge joining x to y. Let X be a subset of V G . Research supported partially by the exchange program between Chinese Academy of Sciences and Japan Society for Promotion of Sciences and by National Natural Science Foundation of China. THE ELECTRONIC .JOURNAL OF COmBINATORICS 6 1999 R4 2 We write G X for the subgraph of G induced by X and dehne X V G n X. The subset X is said to be independent if no two vertices of X are adjacent in G. Sometimes x is used for a singleton x . For a vertex x of G we denote by dG x the degree of x in G that is the number of edges of G incident with x. We denote by Ổ G the minimum degree of G. For integers a and b 0 a b an a b -factor of G is dehned to be a spanning subgraph F of G such that a dF x b for all x 2 V G and an a a -factor is abbreviated to an a-factor. A subset M of E G is called a matching if no two edges of M are adjacent in G. For two graphs H and K the union H u K is the graph with vertex set V H u V K and edge set E H u E K and