Induced subgraphs of a tree with constraint degree

For an integer k≥2, a vertex partition (V1,…,Vs) of a graph G is called a k-good partition if dG[Vi](v)≡1 (mod k) for each v∈Vi, i∈{1,…,s}. We characterize all trees with a k-good partition. Let f1,k(G)=max{|V(H)|: H is an induced subgraph H of G with dH(v)≡1 (mod k) for every vertex v}. In 1997, Berman et al. (1997) showed that f1,k(T)≥2⌊n+2k−32k−1⌋ for any tree T of order n. By sacrificing the bound slightly, but with a much simpler way, we are able to show that f1,k(T)≥nk, with equality if and only if T is the balanced double star of order 2k. Let f0,k1(T) be the maximum cardinality of a subset S⊆V(T) with dT[S](v)=1 or dT[S](v)≡0 (mod k) for each v∈S. In addition, we give a short proof of the result, due to Huang and Hou [9], that f0,k1(T)≥2n3 for any tree T and integer k≥3.