On restricted matching extension in planar graphs (Q5937576)

A graph \(G\) has property \(E(m,n)\) if for each pair of disjoint matchings \(M\) and \(N\) of respective sizes \(m\) and \(n\) there is a perfect matching in \(G\) which includes \(M\) but uses no edge of \(N\). Assume that \(G\) is planar and has an even number of vertices. The authors show that \(G\) does not have \(E(2,1)\). However, if \(G\) is 4-connected then it has \(E(1,1)\) and \(E(0,3)\), while if \(G\) is 5-connected then it has \(E(1,2)\) and \(E(0,4)\). Examples are given to prove these results are sharp, although it is sometimes left to the reader to establish that they have the properties claimed. Two examples for which proofs are given are (i) some 3-connected planar graphs do not have \(E(0,0)\), meaning they do not have a perfect matching, and (ii) for any \(n\) there are planar graphs which have \(E(1,n)\).