The average size of maximal matchings in graphs

Pub Date : 2024-04-04 DOI:10.1007/s10878-024-01144-8
Alain Hertz, Sébastien Bonte, Gauvain Devillez, Hadrien Mélot
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Abstract

We investigate the ratio \(\mathcal {I}(G)\) of the average size of a maximal matching to the size of a maximum matching in a graph G. If many maximal matchings have a size close to \(\nu (G)\), this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, \(\mathcal {I}(G)\) approaches \(\frac{1}{2}\). We propose a general technique to determine the asymptotic behavior of \(\mathcal {I}(G)\) for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of \(\mathcal {I}(G)\) which were typically obtained using generating functions, and we then determine the asymptotic value of \(\mathcal {I}(G)\) for other families of graphs, highlighting the spectrum of possible values of this graph invariant between \(\frac{1}{2}\) and 1.

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图中最大匹配的平均大小
我们研究了图 G 中最大匹配的平均大小与最大匹配的大小之比 \(\mathcal{I}(G)\)。如果许多最大匹配的大小接近 \(\nu(G)\),那么这个图不变式的值就接近 1。相反,如果许多最大匹配的大小很小,那么(\mathcal {I}(G)\) 接近(\frac{1}{2}\)。我们提出了一种通用技术来确定各种图类的\(\mathcal {I}(G)\) 的渐近行为。为了说明这一技术的使用,我们首先展示了它是如何使我们找到已知的 \(\mathcal {I}(G)\) 的渐近值成为可能的,这些值通常是通过生成函数得到的,然后我们确定了 \(\mathcal {I}(G)\) 对于其他图形族的渐近值,强调了这个图形不变式的可能值在\(\frac{1}{2}\) 和 1 之间的频谱。
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