A Critical Assessment of Graceful Graphs and Trees: Study and Development of New Approaches

Abstract

AgraphGisgracefulifitssetofverticesisdenotedbytheset{0,1,...,m}andthe set of edges is denoted by {1,2,...,m}, such that all edges are labeled uniquely and accordingtothedifferenceofthelabelsoftheverticesconnectingit. Asimplenotation to be mentioned that all the vertices will have unique labels, and so will the edges. The idea of graceful labeling came to being from Ringel’s conjecture(see chapter 1). The conjecture introduced in 1963, established the open problem of graceful labeling. Any graphthatcanbelabeledgracefullysuggeststhattheclassofthegraphimpliesRingel’s conjecture. Over the years many people worked on this conjecture and found out many classesofgracefultreesandgraph. SuchgracefulclassesoftreesandgraphsarePathor Chain, Caterpillar, Extended caterpillar, super caterpillar, Star, Olive tree, Banana tree, Lobstar, product tree Cyle wheel, Crown graph etc. A C4 graph is a cycle consisting of four vertices. The C4 graph is graceful. However, graphs formed by multiple number of C4 cycles in them were yet to be studied. Another type of graph is a star. A star S1, nisacentervertexconnectedwithnleaves. Astarisalsoclassifiedasatree,according to the definition. It has been proved earlier that all stars are graceful. The purpose of this paper was to study and develop procedures to gracefully labeled graphs formed by thecombinationofmultiple C4 cyclesandthecombinationofmultiplestars. Ageneral procedure was developed for each of the three classes, proved to be graceful, through thispaper. Furtherresearchwasconductedonfindingmorevarietiesofsuchgraphsand the results were discussed. The result of the research proposes an open problem about aclassofgraphswhichpartiallysupportsRingel’sconjecture. Thepaperhighlightsthe classes of graphs and trees that were previously proved to be graceful.