European Journal of Combinatorics ( IF 1 ) Pub Date : 2020-12-10 , DOI: 10.1016/j.ejc.2020.103278 Michael Fuchs , Guan-Ru Yu , Louxin Zhang
In a recent paper, McDiarmid, Semple, and Welsh (2015) showed that the number of tree-child networks with leaves has the factor in its main asymptotic growth term. In this paper, we improve this by completely identifying the main asymptotic growth term up to a constant. More precisely, we show that the number of tree-child networks with leaves grows like where is the largest root of the Airy function of the first kind. For the proof, we bijectively map the underlying graph-theoretical problem onto a problem on words. For the latter, we can find a recurrence to which a recent powerful asymptotic method of Elvey Price, Fang, and Wallner (2019) can be applied.
中文翻译:
关于树子网络数量的渐近增长
McDiarmid,Semple和Welsh(2015)在最近的一篇论文中指出, 叶有因素 在其主要的渐近增长期。在本文中,我们通过完全确定一个主要的渐近增长项来改善这一点。更确切地说,我们证明了具有 叶子长得像 哪里 是第一类Airy函数的最大根源。为了证明这一点,我们将基础的图论问题双射映射到单词问题上。对于后者,我们可以找到一种递归,可以应用Elvey Price,Fang和Wallner(2019)的一种最新的强大渐近方法。