A tight upper bound on the average order of dominating sets of a graph

Pub Date : 2024-06-17 DOI:10.1002/jgt.23143

Iain Beaton, Ben Cameron

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Abstract

In this paper we study the average order of dominating sets in a graph, $avd (G)$ . Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs $G$ of order $n$ without isolated vertices, $avd (G) \leq 2 n / 3$ . Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have $avd (G) = 2 n / 3$ . We also use our bounds to prove an average version of Vizing's conjecture.

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图的支配集平均阶数的严格上限

本文研究图中支配集的平均阶数 avd ( G ) $\,\text{avd}\,(G)$ 。与其他平均图参数一样，极值图也很值得关注。比顿和布朗猜想，对于所有阶数为 n $n$ 的无孤立顶点的图 G $G$ ，avd ( G ) ≤ 2 n / 3 $,\text{avd}\,(G)\le 2n/3$ 。最近，埃雷证明了无孤立顶点森林的猜想。在本文中，我们证明了这个猜想，并分类了哪些图具有 avd ( G ) = 2 n / 3 $\,\text{avd}\,(G)=2n/3$ 。我们还利用我们的边界证明了平均版本的 Vizing 猜想。

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