A digraph is {\bf 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. A digraph is {\bf semicomplete} if it has no pair of non-adjacent vertices. A {\bf tournament} is a semicomplete digraph without directed cycles of length 2. Fraise and Thomassen \cite{fraisseGC3} proved that every $(k+1)$-strong tournament has a hamiltonian cycle which avoids any prescribed set of $k$ arcs. In \cite{bangsupereuler} the authors demonstrated that a number of results concerning vertex-connectivity and hamiltonian cycles in tournaments and have analogues when we replace vertex connectivity by arc-connectivity and hamiltonian cycles by spanning eulerian subdigraphs. They showed the existence of a smallest function $f(k)$ such that every $f(k)$-arc-strong semicomplete digraph has a spanning eulerian subdigraph which avoids any prescribed set of $k$ arcs. They proved that $f(k)\leq \frac{(k+1)^2}{4}+1$ and also proved that $f(k)=k+1$ when $k=2,3$. Based on this they conjectured that $f(k)=k+1$ for all $k\geq 0$. In this paper we prove that $f(k)\leq (\lceil\frac{6k+1}{5}\rceil)$.

中文翻译：

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跨越欧拉子图避免锦标赛中的 k 个规定弧

一个有向图是 {\bf eulerian} 如果它是连通的并且每个顶点的入度都等于它的出度。因此，具有跨越的欧拉子图是具有汉密尔顿循环的弱化。如果一个有向图没有一对不相邻的顶点，则它是 {\bf semicomplete}。{\bf 锦标赛} 是一个没有长度为 2 的有向循环的半完全有向图。 Fraise 和 Thomassen \cite{fraisseGC3} 证明每个 $(k+1)$-strong 锦标赛都有一个汉密尔顿循环，它避免了任何规定的 $k 集合$ 弧。在 \cite{bangsupereuler} 中，作者证明了一些关于锦标赛中的顶点连通性和哈密顿循环的结果，并且当我们用弧连通性替换顶点连通性和通过跨越欧拉子图来替换哈密顿循环时有类似的结果。他们证明了最小函数 $f(k)$ 的存在，使得每个 $f(k)$-arc-strong 半完全有向图都有一个跨越欧拉子有向图，它避免了任何规定的 $k$ 弧集。他们证明了 $f(k)\leq \frac{(k+1)^2}{4}+1$ 并且还证明了 $f(k)=k+1$ 当 $k=2,3$ 时。基于此，他们推测所有 $k\geq 0$ 的 $f(k)=k+1$。在本文中，我们证明了 $f(k)\leq (\lceil\frac{6k+1}{5}\rceil)$。