Contractible Edges and Removable Edges in 3-Connected Graphs

Abstract

In this paper we prove that every spanning tree of a 3-connected 3-regular graph, except for $$K_4$$ and $$K_2\, \Box \,K_3$$, contains at least two contractible edges, and that every spanning tree of a minimally 3-connected graph, except for the wheel graphs, contains at least one contractible edge. Moreover, we show that every maximum matching of a 3-connected graph of order at least 6 that does not contain the maximal semiwheel graphs avoids at least four removable edges; every maximum matching of a 3-connected graph avoids at least two removable edges.