Matching extension in toroidal quadrangulations II: the 3-extendable case.
Robert E. L. Aldred, Qiuli Li, Michael D. Plummer, Dong Ye, Heping Zhang · Australas. J Comb. · 2015
A graph G containing a perfect matching is said to be m-extendable if m ≤ (|V (G)| − 2)/2 and for every matching M with |M | = m, there is a perfect matching F in G such that M ⊆ F . In a previous paper, four of the present five authors characterized those quadrangulations of the torus which are 2-extendable. In the present work a characterization of those which are 3-extendable is obtained. Since no quadrangulation of the torus can be m-extendable for any m ≥ 4, this completes the study of m-extendability for toroidal quadrangulations. Moreover, by another previous result, it follows that we have therefore characterized all 3-extendable toroidal graphs.