Proper Disconnection of Graphs

Abstract

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