Spanning eulerian subdigraphs in semicomplete digraphs

A 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 (D,a) $(D,a)$ of a semicomplete digraph D $D$ and an arc a $a$ such that D $D$ has a spanning eulerian subdigraph containing a $a$. In particular, we show that if D $D$ is 2-arc-strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs ( D , a ) $(D,a)$ of a semicomplete digraph D $D$ and an arc a $a$ such that D $D$ has a spanning eulerian subdigraph avoiding a $a$. 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 f ( k ) $f(k)$ such that every f ( k ) $f(k)$-arc-strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of k $k$ arcs. We conjecture that f ( k ) = k + 1 $f(k)=k+1$ and establish this conjecture for k <= 3 $k\le 3$ and when the k $k$ arcs that we delete form a forest of stars. A digraph D $D$ is eulerian-connected if for any two distinct vertices x , y $x,y$, the digraph D $D$ has a spanning ( x , y ) $(x,y)$-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.