The study determines edge limits for non-trivial graphs with specific matching criteria, confirming a conjecture.
Let be three integers and be a sufficiently small positive number such that , , and . A ‐graph is called non‐trivial if it has no isolated vertex. In this paper, we determine the maximum number of edges in a non‐trivial ‐graph with vertices and matching number at most . This result confirms a conjecture proposed by Frankl (On non‐trivial families without a perfect matching, European J. Combin ., 84 (2020), 103044) for the case when is sufficiently large.
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Guo et al. (2026) studied this question.
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