Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Pan-factorial Property in Regular Graphs. | Pan-factorial Property in Regular Graphs M. Kano1 and Qinglin Yu23 1 Department of Compuetr and Information Sciences Ibaraki University Hitachi Ibaraki 316-8511 Japan kano@ 2Center for Combinatorics LPMC Nankai University Tianjing PR China yu@ 3Department of Mathematics and Statistics Thompson Rivers University Kamloops BC Canada yu@ Abstract Among other results we show that if for any given edge e of an r-regular graph G of even order G has a 1-factor containing e then G has a fc-factor containing e and another one avoiding e for all k 1 k r 1. Submitted Nov 4 2004 Accepted Nov 7 2005 Published Nov 15 2005 MSC 05C70 05C75. Keywords pan-factorial property 1-factor fc-factor. For a function f V G 0 1 2 3 . a spanning subgraph F of G with degF x f x for all x 2 V G is called an f-factor of G where degF x denotes the degree of x in F. If f x k for all vertices x 2 V G then an f-factor is also called a k-regular factor or a k-factor. An a b -factor is a spanning subgraph F of G such that a degF x b for all x 2 V G . A graph G is pan-factorial if G contains all fc-factors for 1 k 5 G . In this note we investigate the pan-factor property in regular graphs. Moreover we proved that the existence of 1-factor containing any given edge implies the existence of fc-factors containing or avoiding any given edge. The first of our main results is the following. Authors would like to thank the support from the National Science Foundation of China and the Natural Sciences and Engineering Research Council of Canada THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N23 1 Theorem 1 Let G be a connected r-regular graph of even order. If for every edge e of G G has a 1-factor containing e then G has a k-factor containing e and another k-factor avoiding e for all integers k 1 k r 1. The next theorem is also one of our main results. Theorem 2 Let G be a connected graph of even order e be an edge of G and a b c be odd integers such that 1 a c b. If G has .
