Spanning eulerian subdigraphs in semicomplete digraphs
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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) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph containing a. In particular, we show that if D is 2-arc-strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs (D,a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph avoiding 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) such that every f(k)-arc-strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of k arcs: we prove f(k)≤ (k+1)^2/4 +1, conjecture f(k)=k+1 and establish this conjecture for k≤ 3 and when the k arcs that we delete form a forest of stars. A digraph D is eulerian-connected if for any two distinct vertices x,y, the digraph D has a spanning (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 cycles in semicomplete digraphs.
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