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A path in an edge-colored graph is called a monochromatic path if all the edges on the path are colored with one same color. An edge-coloring of G is a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices in G. For a connected graph G, the monochromatic connection number of G, denoted by mc(G), is defined to be the maximum number of colors used in an MC-coloring of G. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we study two kinds of Erdős-Gallai-type problems for mc(G), and completely solve them.

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A tree in an edge-colored graph G is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers k, with k ≥ 3, the (k, )-rainbow index r x k, (G) of G is the minimum number of colors needed in an edge-coloring of G such that for any set S of k vertices of G, there exist internally disjoint rainbow trees connecting S. This concept was introduced by Chartrand et. al., and there have been very few known results about it. In this paper, we establish a sharp threshold function for r x k,