图的长控制圈(英文)

LONG DOMINATING CYCLES IN GRAPHS

设G是具有一个控制圈的图 ,证明了如果在G的每一个圈C上总存在点ν0 ,使得dR(ν0 ) >1,其中R =V(G) \V(C) ,那么G必包含一个长度至少为min{n ,2NC2 (G) -1}的控制圈 ;如果G的每一个控制圈为偶圈 ,那么 ,G包含一个长度为min{n ,2NC2 (G) }的控制圈 ,从而证明了R .Shen和F .Tian的猜想 .

In this paper,we prove that if G contains a dominating cycle and δ ≥2 with a vertex ν 0∈V(G) such that d R(ν 0)>1 ,where R=V(G)\V(C) for all of dominating cycle C , then G contains a dominating cycle of length at least min {n,2NC2(G)-1} ,and if every D -cycle is even cycle then G contains a dominating cycle of length at least min {n,2NC2(G)} ,which locally proves conjectures 1 and 2 of R.Shen and F.Tian.