Distance Matching Extension in Cubic Bipartite Graphs

Abstract

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 $e \in E(G)$, there exist two cycles $C_1$, $C_2$ of length at most d such that $E(C_1) \cap E(C_2) = \{e\}$, then G is distance $d-1$ matchable, and (ii) if G is a planar or projective planar cubic bipartite graph in which, for each $e \in E(G)$, there exist two cycles $C_1$, $C_2$ of length at most 6 such that $e \in E(C_1) \cap E(C_2)$, then G is distance 6 matchable.