摘要
Ｎｏｗｓｔｕｄｙｉｎｇｆｏｒｄｏｃｔｏｒ＇ｓｄｅｇｒｅｅａｔｉｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＡｃａｄｅｍｉａＳｉｎｉｃａ．ＴｈｅｏｒｅｍＤＩ４］Ｌｅｔｋ２１ｂｅａｎｉｎｔｅｇｅｒ．ＩｆＧｉｓａｃｏｎｎｅｃｔｅｄｃｌａｗ－ｆｒｅｅｇｍｐｈｗｉｔｈｈｉＶ（Ｇ）ｌｅｖｅｎａｎｄｗｉｔｈｍｉｎｉｍｕｍｄｅｇｒｅｅｅ（Ｇ）ａｔｌｅａｓｔＺｋ，ｔｈｅｎＧｈａｓａｋ－ｆａｃｔｏｒ．Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｇｅｎｅｒａｌｉｚｅｄｔｈｅｒｅｓｕｌｔｏｆＴｈｅｏｒｅｍＣ，ａｎｄｏｂｔａｉｎｔｈｅｆｏｌｌｏｗｉｎｇＴｈｅｏｒｅｍｉｆＧｉｓａｎＮ＇－ｌｏｃａｌｌｙｃｏｎｎｅｃｔｅｄｃｌａｗｔｈｅｅｇｒａｐｈｗｉｔｈｂ（Ｇ）２２，ｔｈｅｎＧｈａｓａ２－ｆａｃｔｏｒ．２．ＬｅｍｍａｓＬｅｍｍａ１ＩｆＧｉｓａｎＮ＇－ｌｏｃａｌｌｙｃｏｎｎｅｃｔｅｄｃｌａｗ－ａｃｅｇａｓｈｗｉｔｈ６（Ｇ）２２，ｔｈｅｎｆｏｒｅａｃｈｘｏ６Ｖ（Ｇ），Ｇｈａｓａｓｈｏｎｅｓｔｃｙｃｌｅｃｏｎｔａｉｎｉｎｇｘｏａｎｄｈａｖｉｎｇａｔｍｏｓｔ５ｖｅｎｉｃｅｓ．Ｌｅｍｍａ２ＩｆＧｉｓａｎＮ＇－－ｌｏｃａｌｌｙｃｏｎｎｅｃｔｅｄｃｌａｗ－ｆｒ?
Let G =(V, E) be a graph,and k>1 an integer. For v ∈ V(G),denote Nk(v) ={x ∈ V(G)| 1< d(x,v) < k}. We call G an Nk-locally connected graph if,for every vertex v ∈ V(G),G[Nk(v)] is connected.We prove in this paper that if G is N2-locally connected and δ(G)>2 then G has a 2-factor.
