The (vertex-)monochromatic index of a graph

Abstract

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