Upper bounds for the $MD$-numbers and characterization of extremal graphs

Abstract

For an edge-colored graph G, we call an edge-cut M of G monochromatic if the edges of M are colored with the same color. The graph G is called monochromatic disconnected if any two distinct vertices of G are separated by a monochromatic edge-cut. For a connected graph G, the monochromatic disconnection number (or M D-number for short) of G, denoted by md(G), is the maximum number of colors that are allowed in order to make G monochromatic disconnected. For graphs with diameter one, they are complete graphs and so their M D-numbers are 1. For graphs with diameter at least 3, we can construct 2-connected graphs such that their M D-numbers can be arbitrarily large; whereas for graphs G with diameter two, we show that if G is a 2-connected graph then md(G) ≤ 2, and if G has a cut-vertex then md(G) is equal to the number of blocks of G. So, we will focus on studying 2-connected graphs with diameter two, and give two upper bounds of their M D-numbers depending on their connectivity and independent numbers, respectively. We also characterize the n 2 -connected graphs (with large connectivity) whose M D-numbers are 2 and the 2-connected graphs (with small connectivity) whose M D-numbers archive the upper bound n 2 . For graphs with connectivity less than n 2 , we show that if the connectivity of a graph is in linear with its order n, then its M D-number is upper bounded by a constant, and this suggests us to leave a conjecture that for a k-connected graph G, md(G) ≤ n k .