New Results on Graph Matching from Degree-Preserving Growth

The recently introduced model in S. R. Kharel et al.’s study [Degree-preserving network growth. Nature Physics 2022, 18, 100–106] uses matchings to insert new vertices of prescribed degrees into the current graph of an ever-growing graph sequence. The process depends both on the size of the largest available matching, which is the focus of this paper, as well as on the actual choice of the matching. Here, we first show that the question of whether a graphic degree sequence, extended with a new degree 2δ, remains graphic is equivalent to the existence of a realization of the original degree sequence with a matching of size δ. Secondly, we present lower bounds for the size of the maximum matchings in any realization of the degree sequence. We then study the bounds on the size of maximal matchings in some realizations of the sequence, known as the potential matching number. We also estimate the minimum size of both maximal and maximum matchings, as determined by the degree sequence, independently of graphical realizations. Along this line we answer a question raised by T. Biedl et al.: Tight bounds on maximal and maximum matchings. Discrete Mathematics 2004, 285, 7–15.