Geodesic
Perfect matchings and $K_{1, p}$-restricted graphs
Diskretnaya Matematika, Tome 32 (2020) no. 1, pp. 27-50

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A graph is called $K_{1, p}$-restricted ($p \ge 3$) if for every vertex of the graph there are at least $p - 2$ edges between any $p$ of its neighbours. We establish sufficient conditions for the existence of a perfect matching in $K_{1, p}$-restricted graphs in terms of their connectivity and vertex degrees. These conditions imply, in particular, the classical Petersen's result: any $2$-edge-connected $3$-regular graph contains a perfect matching.
Keywords: $K_{1, p}$-restricted graph, strongly $K_{1, p}$-restricted graph, perfect matching, factor-critical graph. 
@article{DM_2020_32_1_a2,
     author = {P. A. Irzhavski and Yu. L. Orlovich},
     title = {Perfect matchings and $K_{1, p}$-restricted graphs},
     journal = {Diskretnaya Matematika},
     pages = {27--50},
     publisher = {mathdoc},
     volume = {32},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2020_32_1_a2/}
}
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P. A. Irzhavski; Yu. L. Orlovich. Perfect matchings and $K_{1, p}$-restricted graphs. Diskretnaya Matematika, Tome 32 (2020) no. 1, pp. 27-50. http://geodesic.mathdoc.fr/item/DM_2020_32_1_a2/