Distance matching extension and local structure of graphs

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