All Lobsters with Perfect Matchings are Graceful

Given a tree, T, consider one of its longest paths, PT, not necessarily unique. We define T to be m–distant if no vertices of T are a distance greater than m away from PT. We will show that all 3–distant graphs are graceful, providing they satisfy the following properties. ave perfect matchings. an be constructed by attaching paths of length 2 to the vertices of a 1–distant tree (caterpillar), by the identification of their end vertices. equence of this is that all 2–distant trees (lobsters) having perfect matchings are graceful.