tailieunhanh - Báo cáo toán học: "On k-Ordered Bipartite Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On k-Ordered Bipartite Graphs | On k-Ordered Bipartite Graphs Jill R. Faudree University of Alaska Fairbanks Fairbanks AK 99775 ffjrf@ Florian Pfender Emory University Atlanta GA 30322 fpfende@ Ronald J. Gould Emory University Atlanta GA 30322 rg@ Allison Wolf College of Computing Georgia Institute of Technology Atlanta GA 30332 awolf@ Submitted Oct 30 2001 Accepted Mar 26 2003 Published Apr 15 2003 MSC Subject Classifications 05C35 05C45 Abstract In 1997 Ng and Schultz introduced the idea of cycle orderability. For a positive integer k a graph G is k-ordered if for every ordered sequence of k vertices there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a hamiltonian cycle then G is said to be k-ordered hamiltonian. We give minimum degree conditions and sum of degree conditions for nonadjacent vertices that imply a balanced bipartite graph to be k-ordered hamiltonian. For example let G be a balanced bipartite graph on 2n vertices n sufficiently large. We show that for any positive integer k if the minimum degree of G is at least 2n k 1 4 then G is k-ordered hamiltonian. 1 Introduction Over the years hamiltonian graphs have been widely studied. A variety of related properties have also been considered. Some of the properties are weaker for example traceability in graphs while others are stronger for example hamiltonian connectedness. Recently a new strong hamiltonian property was introduced in 3 . We say a graph G on n vertices n 3 is k-ordered for an integer k 1 k n if for every sequence S x1 x2 . xk of k distinct vertices in G there exists a cycle that contains all the vertices of S in the designated order. A graph is k-ordered hamiltonian if for every sequence S of k vertices there exists a hamiltonian cycle which encounters the vertices in S in the designated order. We will always let S xi x2 . xk denote the ordered k-set. If we say a cycle C contains S we mean C contains S in the .