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 deg(F)(x) = 1 for every red vertex x and deg(F)(y) = 2 for every blue vertex y.