Spanning eulerian subdigraphs in semicomplete digraphs

Abstract

AbstractA digraph is eulerian if it is connected and every vertex has its in‐degree equal to its out‐degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs of a semicomplete digraph and an arc such that has a spanning eulerian subdigraph containing . In particular, we show that if is 2‐arc‐strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs of a semicomplete digraph and an arc such that has a spanning eulerian subdigraph avoiding . In particular, we prove that every 2‐arc‐strong semicomplete digraph has a spanning eulerian subdigraph avoiding any prescribed arc. We also prove the existence of a (minimum) function such that every ‐arc‐strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of arcs. We conjecture that and establish this conjecture for and when the arcs that we delete form a forest of stars. A digraph is eulerian‐connected if for any two distinct vertices , the digraph has a spanning ‐trail. We prove that every 2‐arc‐strong semicomplete digraph is eulerian‐connected. All our results may be seen as arc analogues of well‐known results on hamiltonian paths and cycles in semicomplete digraphs.