More on the Colorful Monochromatic Connectivity

Abstract

An 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-Renyi random graph model, in which each of the \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) pairs of vertices appears as an edge with probability p independent from other pairs. For any function f(n) satisfying \(1\le f(n)<\frac{1}{2}n(n-1)\), we show that if \(\ell n \log n\le f(n)<\frac{1}{2}n(n-1)\), where \(\ell \in \mathbb {R}^+\), then \(p=\frac{f(n)+n\log \log n}{n^2}\) is a sharp threshold function for the property \(mc\left( G\left( n,p\right) \right) \ge f(n)\); if \(f(n)=o(n\log n)\), then \(p=\frac{\log n}{n}\) is a sharp threshold function for the property \(mc\left( G\left( n,p\right) \right) \ge f(n)\).