Distance Matching Extension in Cubic Bipartite Graphs

A graph G is said to be distanced matchable if, for any matching M of G in which edges are pairwise at least distance d apart, there exists a perfect matching M^∗ of G which contains M. In this paper, we prove the following results: (i) if G is a cubic bipartite graph in which, for each e∈ E(G) , there exist two cycles C₁, C₂ of length at most d such that E(C₁) ∩ E(C₂) = { e} , then G is distance d- 1 matchable, and (ii) if G is a planar or projective planar cubic bipartite graph in which, for each e∈ E(G) , there exist two cycles C₁, C₂ of length at most 6 such that e∈ E(C₁) ∩ E(C₂) , then G is distance 6 matchable.