On matching extensions with prescribed and proscribed edge sets. II (Q1292810)

A graph \(G\) is said to be \(E(m,n)\) if for every pair of disjoint matchings \(M,N\subseteq E(G)\) of size \(m\) and \(n\), respectively, there is a perfect matching \(F\) in \(G\) such that \(M\subseteq F\) and \(F\cap N=\varnothing\). The paper studies conditions under which the implication \(E(m,n)\to E(p,q)\) does or does not hold. It extends the previous results on claw-free graphs and bipartite graphs.