On graphic degree sequences and matching numbers

The recently introduced Degree Preserving Growth model (Nature Physics, DOI:10.1038/s41567-021-01417-7) uses matchings to insert new vertices of prescribed degrees into the current graph of an ever-growing graph sequence. In this paper we are presenting lower bounds on the matching numbers of the graphs in the growth sequence, based on their degree sequence only. We are interested in bounds on maximal/maximum matchings in some realizations (potential matching number), and also on common bounds on all realizations (forcible matching number). Particularly, we prove that whether a degree sequence can be extended with a new degree $2\delta$ is equivalent to the existence of a realization of the original degree sequence with a matching of size $\delta$ that covers the largest $2\delta$ degrees. Second, we estimate the minimum size of both the maximal and the maximum matchings, as determined by the degree sequence, independently of the graphical realization. Furthermore, we also estimate the maximum value of the matching number over all possible realizations for the degree sequence. Along this line we answer a question raised by Biedl, Demaine et. al. (DOI:10.1016/j.disc.2004.05.003). We briefly discuss an application to prime gap sequences.
Comment: In the new version we removed a mistaken citation, some unnecessary citations, furthermore some too simple reasoning. 15 pages