Upper bounds for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e55" altimg="si1358.svg"><mml:mrow><mml:mi>M</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:math>-numbers and characterization of extremal graphs

For an edge-colored graph G, we call edge-cut M of G monochromatic if the edges are colored with same color. The is called disconnected any two distinct vertices separated by a edge-cut. connected disconnection number (or MD-number for short) denoted md(G), maximum colors that allowed in order to make disconnected. graphs diameter one, they complete and so their MD-numbers 1. at least 3, can construct 2-connected such be arbitrarily large; whereas two, show then md(G)≤2, has cut-vertex md(G) equal blocks G. So, will focus on studying give upper bounds depending connectivity independent numbers, respectively. We also characterize n2-connected (with large connectivity) whose 2 small achieve bound n2 (these extremal graphs). less than n2, linear its n, bounded constant, this suggests us leave conjecture k-connected md(G)≤nk.