More on the Colorful Monochromatic Connectivity

n edge-coloring of a connected graph is a monochromatically-connecting coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices, which was introduced by Caro and Yuster. Let mc ( G ) denote the maximum number of colors used in an MC-coloring of a graph G . Note that an MC-coloring does not exist if G is not connected, in which case we simply let mc(G)=0 . In this paper, we characterize all connected graphs of size m with mc(G)=1, 2, 3, 4 , m-1 , m-2 and m-3 , respectively. We use G ( n , p ) to denote the Erdős-Rényi random graph model, in which each of the ( [ n; 2 ]) pairs of vertices appears as an edge with probability p independent from other pairs. For any function f ( n ) satisfying 1≤ f(n)<1/2n(n-1) , we show that if ℓ n log n≤ f(n)<1/2n(n-1) , where ℓ∈ℝ^+ , then p=f(n)+nloglog n/n^2 is a sharp threshold function for the property mc( G( n,p) ) ≥ f(n) ; if f(n)=o(nlog n) , then p=log n/n is a sharp threshold function for the property mc( G( n,p) ) ≥ f(n) .