Tree berdiameter lima adalah graceful

Abstract

This manuscript presents some proofs that every tree of diameter five is graceful. A graceful tree is a graph which has a graceful labelling. This labelling assigns absolute difference of connected vertices, such that the resulting edge labels are mutually distinct. There are three theorems to be considered. First theorem proves that every tree of diameter four is graceful. Second theorem proves that every tree of diameter five is graceful or nearly graceful. The third theorem improves the second theorem and shows that every tree of diameter five is nearly graceful, which can be transformed into graceful using some certain patterns.