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arXiv · 2608.22498

Domination versus edge domination in regular graphs of degree at least seven

Chakshu Gupta
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Abstract

Baste et al. (2020) conjectured that every regular graph of positive degree has domination number at most its edge domination number, the least size of a maximal matching. Combining published bounds settles the inequality for every degree at least nine. A reduction proves the inequality whenever one endpoint of each edge of a minimum maximal matching can be chosen to form a dominating set, and the Lov\'asz Local Lemma shows such a choice exists for every degree at least seven, newly closing degrees seven and eight and leaving degrees three through six open. The reduction settles each open degree up to a bounded number of vertices, forty-eight for cubic graphs. At fifty vertices, however, the reduction meets an explicit cubic graph it cannot settle, though the inequality holds there too. The inequality cannot be tightened, since infinitely many cubic graphs have equal domination and edge domination numbers. The cubic case stays open, and even linear arguments from the local structure cannot close it. The middle degrees stay open beyond the graphs already settled.

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Chakshu Gupta. 2026-08-23. Domination versus edge domination in regular graphs of degree at least seven. https://arxiv.org/abs/2608.22498

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