Distance Matching Extension in Cubic Bipartite Graphs

A graph G is said to be distance d 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 is an element of E(G), there exist two cycles C-1, C-2 of length at most d such that E(C-1) boolean AND E(C-2) = {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 is an element of E(G), there exist two cycles C-1, C-2 of length at most 6 such that e is an element of E(C-1) boolean AND E(C-2), then G is distance 6 matchable.