Factors with Red–Blue Coloring of Claw-Free Graphs and Cubic Graphs

Among some results, we prove the following two theorems. (i) Let G be a connected claw-free graph. We arbitrarily color every vertex of G red or blue so that the number of red vertices is even. Then G has vertex-disjoint paths whose end-vertices are exactly the same as the red vertices of G. (ii) Let G be a 3-edge connected claw-free cubic graph. We arbitrarily color every vertex of G red or blue so that the number of red vertices is even and the distance between any two red vertices is at least 3. Then G has a factor F such that degF(x)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\deg _F(x) =1$$\end{document} for every red vertex x and degF(y)=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\deg _F(y)=2$$\end{document} for every blue vertex y.