A lower bound for the connectivity of directed Euler tour transformation graphs (Q1356528)

Let \(D\) be a directed Eulerian multigraph and \(v\) a vertex of \(D\). The directed Euler tour graph of \(D\), denoted by \(E_u(D)\), is an undirected simple graph defined as follows: The vertices of \(E_u(D)\) are directed Euler tours of \(D\), and two directed Euler tours \(E\) and \(F\) are adjacent in \(E_u(D)\) if they can be obtained from each other by a \(T\)-transformation. X. Xia proved that each directed Euler tour \((T)\)-transformation graph \(E_u (D)\) is connected. The author proves the lower bound \[ \sum_{v\in Q} (d_v-1) (d_v-2)/2 \] for the connectivity of \(E_u(D)\), where \(Q= \{v\in V(D) \mid d_v\geq 2\}\). The author also notes that this lower bound is best possible.