Abstract
Let \({\mathcal{F}}\) be a family of connected graphs. A graph G is said to be \({\mathcal{F}}\)-free if G is H-free for every graph H in \({\mathcal{F}}\). We study the problem of characterizing the families of graphs \({\mathcal{F}}\) such that every large enough connected \({\mathcal{F}}\)-free graph of even order has a perfect matching. This problems was previously studied in Plummer and Saito (J Graph Theory 50(1):1–12, 2005), Fujita et al. (J Combin Theory Ser B 96(3):315–324, 2006) and Ota et al. (J Graph Theory, 67(3):250–259, 2011), where the authors were able to characterize such graph families \({\mathcal{F}}\) restricted to the cases \({|\mathcal{F}|\leq 1, |\mathcal{F}| \leq 2}\) and \({|\mathcal{F}| \leq 3}\), respectively. In this paper, we complete the characterization of all the families that satisfy the above mentioned property. Additionally, we show the families that one gets when adding the condition \({|\mathcal{F}| \leq k}\) for some k ≥ 4.
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Ota, K., Sueiro, G. Forbidden Induced Subgraphs for Perfect Matchings. Graphs and Combinatorics 29, 289–299 (2013). https://doi.org/10.1007/s00373-011-1102-6
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DOI: https://doi.org/10.1007/s00373-011-1102-6