Edge proximity conditions for extendability in regular bipartite graphs

Let m and r be positive integers with r >= 3, let G be an r-regular cyclically ((m - 1) r + 1)-edge-connected bipartite graph and let M be a matching of size m in G. Plummer showed that whenever r >= m + 1, there is a perfect matching of G containing M. When r = 3, Aldred and Jackson, extended this result to the case when m + 1 >= r = 3 by showing there is a perfect matching in G containing M whenever the edges in M are pairwise at least distance f (m) apart where f(m) = { 1, m = 2, 3, 3 <= m <= 4, 4, 5 <= m <= 8 5, m >= 9. In this paper, we relax the condition that r = 3 and the distance restriction introduced by Aldred and Jackson to show that, for m >= r >= 3 and G an r-regular cyclically ((m - 1)r + 1)-edge-connected bipartite graph, for each matching M in G with |M| = m and such that each pair of edges in M is distance at least 3 apart, there is a perfect matching in G containing M.