On certain classes of graceful lobsters

TL;DR

This paper explores classes of graceful lobsters, a type of tree, by developing methods to join graceful graphs and applying these to identify new graceful lobsters, advancing understanding of the Ringel-Kotzig conjecture.

Contribution

It introduces methods for constructing graceful lobsters from known graceful graphs using adjacency matrix techniques, expanding the classes of known graceful trees.

Findings

Identifies new classes of graceful lobsters.

Provides methods for joining graceful graphs.

Shows how to generate more graceful lobsters.

Abstract

A graph G=(V,E) with m edges is graceful if it has a distinct vertex labeling f, a map from V into the set{0,1,2,3,...,m} which induces a distinct edge labeling |f(u)-f(v)| for edges uv in E. The famous Ringel-Kotzig conjecture (1964) is that all trees are graceful. The base of a tree T is obtained from T by deleting its one-degree vertices. A caterpillar is a tree whose base is a path and a lobster is a tree whose base is a caterpillar. Paths and caterpillars are known to be graceful. Next it was conjectured by Bermond (1979) that all lobsters are graceful. In this paper we describe various methods of joining graceful graphs and \alpha-labeled graphs using the adjacency matrix characterization that initiated by Bloom (1979) and others. We apply these results to obtain some classes of graceful lobsters and indicate how to obtain some others.