The Colored Disconnection Numbers of Cellular and Grid Networks

Abstract

Let G be a nontrivial link-colored connected network. A link-cut R of G is called a rainbow link-cut if no two of its links are colored the same. A link-colored network G is rainbow disconnected if for every two nodes u and v of G, there exists a u-v rainbow link-cut separating them. Such a link coloring is called a rainbow disconnection coloring of G. For a connected network G, the rainbow disconnection number of G, denoted by rd(G), is defined as the smallest number of colors that are needed in order to make G rainbow disconnected. Similarly, there are some other new concepts of network colorings, such as proper disconnection coloring, monochromatic disconnection coloring and rainbow node-disconnection coloring. In this paper, we obtain the exact values of the rainbow (node-)disconnection numbers, proper and monochromatic disconnection numbers of cellular networks and grid networks, respectively.