Perfect matchings in highly cyclically connected regular graphs
A leaf matching operation on a graph consists of removing a vertex of degree 1 together with its neighbour from the graph. For k≥ 0, let G be a d-regular cyclically (d-1+2k)-edge-connected graph of even order. We prove that for any given set X of d-1+k edges, there is no 1-factor of G avoiding X if and only if either an isolated vertex can be obtained by a series of leaf matching operations in G-X, or G-X has an independent set that contains more than half of the vertices of G. To demonstrate
... w to check the conditions of the theorem we prove several statements on 2-factors of cubic graphs. For k≥ 3, we prove that given a cubic cyclically (4k-5)-edge-connected graph G and three paths of length k such that the distance of any two of them is at least 8k-17, there is a 2-factor of G that contains one of the paths . We provide a similar statement for two paths when k=3 and k=4. As a corollary we show that given a vertex v in a cyclically 7-edge-connected cubic graph, there is a 2-factor such that v is in a circuit of length greater than 7.