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- W3134389690 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 MD-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 MD-numbers are 1. For graphs with diameter at least 3, we can construct 2-connected graphs such that their MD-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 MD-numbers depending on their connectivity and independent numbers, respectively. We also characterize the n2-connected graphs (with large connectivity) whose MD-numbers are 2 and the 2-connected graphs (with small connectivity) whose MD-numbers achieve the upper bound n2 (these graphs are called extremal graphs). For graphs with connectivity less than n2, we show that if the connectivity of a graph is linear in its order n, then its MD-number is upper bounded by a constant, and this suggests us to leave a conjecture that for a k-connected graph G, md(G)≤nk." @default.
- W3134389690 created "2021-03-15" @default.
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- W3134389690 date "2021-05-01" @default.
- W3134389690 modified "2023-10-16" @default.
- W3134389690 title "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" @default.
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- W3134389690 doi "https://doi.org/10.1016/j.dam.2021.02.023" @default.
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