The (vertex-)monochromatic index of a graph

A tree T in an edge-colored (vertex-colored) graph H is called a monochromatic (vertex-monochromatic) tree if all the edges (internal vertices) of T have the same color. For $$S\subseteq V(H)$$S⊆V(H), a monochromatic (vertex-monochromatic) S-tree in H is a monochromatic (vertex-monochromatic) tree of H containing the vertices of S. For a connected graph G and a given integer k with $$2\le k\le |V(G)|$$2≤k≤|V(G)|, the k-monochromatic index$$mx_k(G)$$mxk(G) (k-monochromatic vertex-index$$mvx_k(G)$$mvxk(G)) of G is the maximum number of colors needed such that for each subset $$S\subseteq V(G)$$S⊆V(G) of k vertices, there exists a monochromatic (vertex-monochromatic) S-tree. For $$k=2$$k=2, Caro and Yuster showed that $$mc(G)=mx_2(G)=|E(G)|-|V(G)|+2$$mc(G)=mx2(G)=|E(G)|-|V(G)|+2 for many graphs, but it is not true in general. In this paper, we show that for $$k\ge 3$$kź3, $$mx_k(G)=|E(G)|-|V(G)|+2$$mxk(G)=|E(G)|-|V(G)|+2 holds for any connected graph G, completely determining the value. However, for the vertex-version $$mvx_k(G)$$mvxk(G) things will change tremendously. We show that for a given connected graph G, and a positive integer L with $$L\le |V(G)|$$L≤|V(G)|, to decide whether $$mvx_k(G)\ge L$$mvxk(G)źL is NP-complete for each integer k such that $$2\le k\le |V(G)|$$2≤k≤|V(G)|. Finally, we obtain some Nordhaus---Gaddum-type results for the k-monochromatic vertex-index.