Proper Disconnection of Graphs

For an edge-colored graph G, a set F of edges G is called proper edge-cut if and any pair adjacent in are assigned different colors. An disconnected for each distinct vertices there exists separating them. connected the disconnection number denoted by pd(G), minimum colors that needed order to make disconnected. In this paper, we first give exact values numbers some special families graphs. Next, obtain sharp upper bound pd(G) n, i.e, \(pd(G)\le \min \{ \chi '(G)-1, \left\lceil \frac{n}{2} \right\rceil \}\). Finally, show given integers k size n with \(pd(G)=k\) \(n-1\) \(k=1\) \(n+2k-4\) \(2\le k\le \lceil \frac{n}{2}\rceil \).