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The smallest number of vertices in a 2-arc-strong digraph which has no good pair

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 Added by Zhenyu Taoqiu
 Publication date 2020
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and research's language is English




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Bang-Jensen, Bessy, Havet and Yeo showed that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching $B^+$ and an in-branching $B^-$ which are arc-disjoint (such two branchings are called a {it good pair}), which settled a conjecture of Thomassen for digraphs of independence number 2. They also proved that every digraph on at most 6 vertices and arc-connectivity at least 2 has a good pair and gave an example of a 2-arc-strong digraph $D$ on 10 vertices with independence number 4 that has no good pair. They asked for the smallest number $n$ of vertices in a 2-arc-strong digraph which has no good pair. In this paper, we prove that every digraph on at most 9 vertices and arc-connectivity at least 2 has a good pair, which solves this problem.



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Hakimi and Schmeichel determined a sharp lower bound for the number of cycles of length 4 in a maximal planar graph with $n$ vertices, $ngeq 5$. It has been shown that the bound is sharp for $n = 5,12$ and $ngeq 14$ vertices. However, it was only conjectured by the authors about the minimum number of cycles of length 4 for maximal planar graphs with the remaining small vertex numbers. In this note we confirm their conjecture.
We prove that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching $B^+$ and an in-branching $B^-$ which are arc-disjoint (we call such branchings good pair). This is best possible in terms of the arc-connectivity as there are infinitely many strong digraphs with independence number 2 and arbitrarily high minimum in-and out-degrees that have good no pair. The result settles a conjecture by Thomassen for digraphs of independence number 2. We prove that every digraph on at most 6 vertices and arc-connectivity at least 2 has a good pair and give an example of a 2-arc-strong digraph $D$ on 10 vertices with independence number 4 that has no good pair. We also show that there are infinitely many digraphs with independence number 7 and arc-connectivity 2 that have no good pair. Finally we pose a number of open problems.
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