Abstract
A digraph D is supereulerian if D has a spanning eulerian subdigraph, or equivalently, a spanning closed 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 k(D) denote the number of connected symmetric components of J. We investigate conditions to assure that D has a spanning trail, and find a family of well-characterized nonsupereulerian digraphs H such that a strong digraph D with k(D)≤3 has a spanning closed trail if and only if D∉H. The main results are applied to obtain a tight bound on the size of a semicomplete bipartite digraph to be supereulerian, with all the extremal digraphs characterized.
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