On saturation problems for matchings with regularity constraints

A graph G is F -saturated if G is F -free but for any edge e in the complement of G the graph G+e contains F . Gerbner et al. (Discrete Math., 345 (2022), 112921) initiated the study of rsat(n,F), the minimum number of edges in a regular n -vertex F -saturated graph. They posed the problem of determining for which graphs F rsat(n,F) exists. Writing (m+1)K₂ for a matching of size m+1, we obtain the precise value of rsat(n,(m+1)K₂) for all possible cases. As a natural counterpart, we also determine the maximum number of edges in a regular n -vertex (m+1)K₂-free graph for all m≥1 and n≥2m+2.