Distance matching extension and local structure of graphs

A matching M in a graph G is said to be extendable if there exists a perfect matching of G containing M . Also, M is said to be a distance d matching if the shortest distance between a pair of edges in M is at least d . A graph G is distance d matchable if every distance d matching is extendable in G , regardless of its size. In this paper, we study the class of distance d matchable graphs. In particular, we prove that for every integer k with k ≥ 3 , there exists a positive integer d such that every connected, locally ( k − 1 ) ‐connected K 1 , k ‐free graph of even order is distance d matchable. We also prove that every connected, locally k ‐connected K 1 , k ‐free graph of even order is distance 3 matchable. Furthermore, we make more detailed analysis of K 1 , 4 ‐free graphs and study their distance matching extension properties.