Let G be a graph and let f positive integer-valued function on V(G). In this paper, we show that if for all S⊆V(G), ω(G∖S)<∑v∈S(f(v)−2)+2+ω(G[S]), then has spanning tree T containing an arbitrary given matching such each vertex v, dT(v)≤f(v), where ω(G∖S) denotes the number of components G∖S ω(G[S]) induced subgraph G[S] with set S. This is improvement several results. Next, prove ω(G∖S)≤∑v∈S(f...

Since the proof of a "colorful" version [Caratheodory's theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_%28convex_hull%29) by Bárány in 1982, it has been an important problem to obtain colorful extensions other classical results discrete geometry (for instance Tverberg's theorem). The present paper continues this line research, but context extremal graph theory rather than ...