Research On The Existence Of HW(r,s; C,3)

The Hamilton-Waterloo problem is to determine the existence of a 2-factorization of K n, n = 2 h+ 1 in which r of the 2-factors are isomorphic to a given 2-factor Q and s of the 2-factors are isomorphic to a given 2-factor R, with r + s = h. If the components of Q are cycles of length c1 , , cq and the components of R are cycles of length d1 , , d t(with∑ci =∑d j= 2 h+1), then the corresponding instance of the Hamilton-Waterloo problem is denoted by ( )HW 2 h + 1; r , s ; c1 , , cq ; d1 , ,dt. The special case of the Hamilton-Waterloo problem that we deal with in this paper is the case c1 = = cq = c, d1 = = dt= 3, briefly by HW ( r , s ; c ,3), which in turn implies that the number of vertices must be 2 h + 1 = 6 k+ 3, let c = 6 kg+3, g | 2 k + 1. We write HW ( r , s ; c ,3) as HW r , s ; 6 kg+3,3 . Let I ( n ) = 0,1, , n2?1 and ( )HWg * 6 k + 3 = r | there exsits HW r , s; 6 kg+3,3 , We draw a conclusion that ( ) { } ( )I 4 k + 3 \ 1 ? HWg*n, ( ) { } ( ) ( )*I n \ 1 ? HW3n ? I n, when k≡1 ( mod3);( ) { } ( )I 4 k + 1 \ 1 ? HWg*n, when k≡0 ( mod3) and k≠3,6.