Summary: A digraph \(D\) is supereulerian if \(D\) has a spanning closed trail, and is strongly trail-connected if for any pair of vertices \(u, v \in V(D)\), \(D\) has a spanning \((u, v)\)-trail and a spanning \((v, u)\)-trail. The symmetric core \(J = J(D)\) of a digraph \(D\) is a spanning subdigraph of \(D\) with \(A(J)\) consisting of all symmetric arcs in \(D\). Let \(J_1, J_2, \dots, J_{k ( D )}\) be the connected symmetric components of \(J\) and define \(\lambda_0(D) = \min \{\lambda( J_i) : 1 \leq i \leq k(D) \}\). We prove that the contraction \(D^\prime = D / J\) can be used to predict the existence of spanning trails in \(D\). It is known that if \(k(D) \leq 2\), then \(D\) has a spanning closed trail. In particular, each of the following holds for a strong digraph \(D\) with \(k(D) \geq 3\).
i) If \(\lambda_0(D) \geq k(D) - 2\), then \(D\) has a spanning trail if and only if \(D^\prime\) has a spanning trail.
ii) If \(\lambda_0(D) \geq k(D) - 1\), then \(D\) is supereulerian if and only if \(D^\prime\) is supereulerian.
iii) If \(\lambda_0(D) \geq k(D)\), then \(D\) is strongly trail-connected if and only if \(D^\prime\) is strongly trail-connected.