Let　G　be　a　graph　which　may　have　multiple　edges　but　no　loops,　mu　G(v)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mu　_{G}(v)$$\end{document}　be　the　multiplicity　of　G　(which　is　the　maximum　among　the　numbers　of　edges　between　v　and　the　other　vertex　on　G),　and　a　and　b　be　any　two　functions,　s.t.　a,b:V(G)-＞　N\{0,1}\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$a,b:　V(G)\rightarrow　\mathbb　{N}　\backslash　\{0,1\}$$\end{document}.　A　theta　is　a　graph　made　of　three　internally　vertex-disjoint　chordless　paths　P1=x-&　ctdot;-y\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$P_{1}=x-　\cdots　-y$$\end{document},　P2=x-&　ctdot;-y\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$P_{2}=x-\cdots　-y$$\end{document},　P3=x-&　ctdot;-y\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$P_{3}=x-\cdots　-y$$\end{document}　of　length　at　least　two　and　no　edges　exist　between　Pi\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$P_i$$\end{document}　and　Pj\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$P_j$$\end{document}　(i　not　equal　j,i,j　is　an　element　of{1,2,3}\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$i\ne　j,i,j\in　\{1,2,3\}$$\end{document})　except　the　three　edges　incident　to　x　and　the　three　edges　incident　to　y.　For　a　partition　(X,　Y)　of　V(G),　and　any　x　is　an　element　of　X\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$x\in　X$$\end{document},　y　is　an　element　of　Y\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$y\in　Y$$\end{document},　let　dX(x)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$d_X(x)$$\end{document}　denote　by　the　degree　of　x　in　G[X],　and　dY(y)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$d_Y(y)$$\end{document}　be　defined　similarly.　In　this　paper,　we　show　that　a　graph　G　admits　a　partition　(A,　B)　such　that　dA(x)＞=　a(x)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$d_{A}(x)\ge　a(x)$$\end{document}　for　any　x　is　an　element　of　A\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$x\in　A$$\end{document}　and　dB(y)＞=　b(y)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$d_{B}(y)　\ge　b(y)$$\end{document}　for　any　y　is　an　element　of　B\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$y\in　B$$\end{document}　if　G　is　H\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\mathcal　{H}$$\end{document}-free　and　dG(v)＞=　a+b+2　mu　G(v)-3\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$d_{G}(v)\ge　a　+　b　+　2\mu　_{G}(v)-3$$\end{document}　for　any　vertex　v　is　an　element　of　V(G)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$v\in　V(G)$$\end{document},　where　for　each　member　H　is　an　element　of　H\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$H　\in　\mathcal　{H}$$\end{document},　the　underlying　of　H　belongs　to　{triangle,　wheel,　theta}.